UNIVERSITY  OF  CALIFORNIA 
LOWER  DIVISIOH 


LIBRARY 

OF  THE 


Received  .....  . 


Accessions  No<L./..  &.  ..........          Book  No  .....  L.. 


SOPHOMORE    COURSE 


PHYSICAL  MEASUREMENTS 


BY 

ELMER  R.   DREW,  B.  S. 

INSTRUCTOR   IN  PHYSICS  IN  THE  UNIVERSITY  OF  CALIFORNIA 


BERKELEY,  CALIFORNIA 
1808 


7 


n 


Entered  According  to  Act  of  Congress  in  the  Year  1898,  by 

ELMER    R.    DREW 

In  the  Office  of  the  Librarian  of  Congress  at  Washington. 


PREFACE. 


THIS  Course  has  been  given  for  the  last  three  years  to  Sophomore 
students  in  the  Colleges  of  Engineering,  Chemistry,  Agriculture  and 
Natural  Sciences,  and  accompanies  a  Lecture  Course  in  General 
Physics.  It  occupies  two  3-hour  periods  per  week  throughout  the 
year,  being  preceded  by  a  Freshman  Course  of  two  laboratory  periods 
and  one  lecture  per  week  for  one  year,  and  by  a  Matriculation  Ex- 
amination in  Physics.  The  students'  preparation,  as  thus  indicated, 
will  account  for  the  fact  that  the  present  work  is  confined  mainly  to 
directions  for  handling  the  apparatus  used  in  the  experiments,  the 
theory  being  either  briefly  formulated  or  entirely  omitted. 

It  is  intended  to  so  write  the  exercises  that  they  can  not  be  carried 
out  without  much  careful  thought  on  the  part  of  the  student.  In 
giving  them  this  form,  two  ideas  have  been  kept  in  mind:  First, 
not  to  require  of  the  student  anything  which  his  preparation  does 
not  warrant;  but  within  this  limit,  to  compel  him  to  build  his  own 
steps,  with  no  more  guidance  than  is  really  necessary.  Second,  to 
take  nothing  on  authority;  but  to  have  worked  out,  in  some  form, 
the  reasoning  underlying  every  method  and  formula  used. 

These  ideas  are  further  carried  out,  and  results  tested,  in  four 
written  examinations  which  are  held  at  convenient  intervals  during 
the  year.  When  the  results  show  it  to  be  necessary,  these  examina- 
tions are  supplemented  by  recitations  and  lectures.  Rather  than 
say  too  much  in  the  directions,  it  is  thought  best  to  say  too  little, 
encouraging  the  student  to  build  up  his  working  knowledge  of  the 
subject  for  himself,  according  to  his  ability,  and  then  to  assist  him 
in  properly  completing  the  structure. 

Little  originality  can  be  claimed  for  the  choice  of  subject-matter 
673151  (iii) 


IV  PREFACE. 

for  the  experiments;  but  it  is  hoped  that  in  the  sequence  in  which 
the  exercises  are  arranged,  and  in  the  methods  of  presenting  the  de- 
tails of  each  experiment,  some  improvement  over  the  usual  texts 
may  be  noted.  The  attempt  has  been  made  to  have  each  exercise 
appear  not  as  an  isolated  piece  of  work,  but  as  part  of  a  larger  whole, 
each  part  depending  more  or  less  upon  what  has  preceded  it. 

In  pursuance  of  this  idea,  no  measurement  has  been  prescribed 
merely  for  practice,  but  always  to  serve  a  definite  purpose  in  obtain- 
ing a  desired  result. 

The  first  two  experiments,  A  and  B,  do  not  form  an  integral  part 
of  the  course,  and  are  assigned  only  in  special  cases,  to  students  who 
are  unable  to  begin  the  regular  sequence  of  experiments,  or  are 
obliged  to  interrupt  it  for  any  reason. 

After  completing  the  exercises  here  laid  out,  about  six  periods 
are  spent  in  what  is  called  "Special  Problem  Work."  Each  student 
is  required  to  select  some  problem,  within  his  capacity  and  the 
facilities  of  the  laboratory,  which  will  occupy  the  required  time. 
The  experimental  details  and  adaptation  of  apparatus  are  to  be  ar- 
ranged by  the  student,  so  far  as  possible.  One  of  the  chief  aims 
of  the  laboratory  course  has  been  to  give  the  power  to  do  independ- 
ent work  of  a  simple  character,  and  this  exercise  is  a  test  of  the 
student's  ability  in  that  direction. 

The  author  is  under  obligations  to  Professor  Slate,  and  all  the 
members  of  the  Physics  Department,  for  advice  and  constant  en- 
couragement during  the  building  up  of  the  Course.  He  is  also  espe- 
cially indebted  to  Mr.  Arthur  Incell  for  skilful  assistance  in  design- 
ing and  constructing  the  necessary  apparatus,  and  to  Dr.  W.  P. 
Boynton  for  numerous  suggestions,  the  result  of  his  experience  in 
directing  the  Course  during  the  past  year. 

E  R   DREW. 
Berkeley,  July,  1898. 


COURSE  IN 


Physical  Measurements 


GENERAL   DIRECTIONS. 

Each  of  the  following  exercises  is  intended  to  occupy  the 
time  of  one  laboratory  period,  unless  otherwise  stated.  There 
will  usually  be  time,  after  completing  the  observations,  to 
calculate  and  write  out  the  results;  but  this  is  not  required. 

Carbon  copies  of  completed  notes  and  results  must  be  handed 
in  within  two  weeks  of  the  date  on  which  the  experiment  was 
performed.  These  papers  will  be  marked  in  the  following  four 
grades:  I,  excellent;  2,  satisfactory;  3,  deficient  in  some  part, 
which  will  usually  be  indicated;  4,  unsatisfactory  as  a  whole. 
Papers  marked  3  or  4  will  be  raised  to  grade  2  if  put  in  satis- 
factory condition  within  one  week  from  date  of  marking; 
unleso  this  involves  repetition  of  the  experiment,  when  more 
time  will  be  allowed.  Carbon  copies  only  of  such  corrections 
will  be  received. 

Papers  not  handed  in  within  the  time  limits  set  above  will 
not  be  marked  higher  than  grade  3,  except  in  special  cases. 

All  observations  should  be  recorded  directly  in  the  note- 
book. If  for  any  reason  it  is  desirable  to  take  them  on  loose 

(5) 


6  GENERAL    DIRECTIONS. 

paper,  they  should  be  copied  at  once  into  the  book.     If  any 
observations  are  rejected,  the  reason  should  be  stated. 

Neatness  and  orderly  arrangement  of  notes  will  be  insisted 
upon.  Tabulate  numerical  data  when  possible.  Always  indi- 
cate the  denominations  of  numbers.  Numerical  computations 
may  be  merely  indicated.  Make  the  explanatory  matter  full 
enough  so  that,  when  taken  in  connection  with  the  directions, 
it  will  make  each  step  intelligible. 

Much  time  will  be  saved  in  numerical  work  by  selecting  the 
method  of  computation  best  suited  to  the  particular  case, 
whether  logarithms,  slide-rule  or  ordinary  arithmetic.  A 
result  obtained  by  one  method  should  always  be  checked,  at 
ItasJ  rou^l^iiy,  .try  -some  other  method. 

..  Nevej*  f^e  ^satisfied  -\vjth  a  single  reading  or  determination  of 
a  quaalky.  •  •  When  circumstances  will  permit,  take  at  least 
three  readings,  varying  the  conditions  and  the  method  of 
reading  as  widely  as  possible. 

When  two  students  are  working  together  on  an  experiment, 
the  work  should  be  so  arranged  that  each  observation  is  taken 
at  least  once,  usually  several  times,  by  each  observer.  This 
not  only  gives  each  student  the  full  benefit  of  the  experiment, 
but  also  tends  to  eliminate  the  effect  of  the  so-called  "personal 
equation."  The  two  series  of  observations  should  be  distin- 
guished by  appropriate  initials. 

In  using  any  piece  of  apparatus,  note  the  number  which 
will  usually  be  found  stamped  on  some  part  of  it.  This  is 
necessary,  because  the  same  piece  may  be  needed  in  a  later 
experiment. 

The  careful  and  thoughtful  work  which  this  course  requires 
can  not  be  well  done  in  a  noisy  room.  The  large  number 
working  in  the  room  at  once,  makes  it  especially  important 
that  each  one  should  carefully  avoid  making  any  unnecessary 
noise,  by  loud  talking  or  otherwise. 

It  is  of  course  expected  that  the  tables  and  apparatus  used 
will  be  left  in  good  order  at  the  close  of  the  exercise. 


B]  MASS   OF   GIVEN    VOLUME   OF   AIR. 


A.    MAGNIFYING   POWER  OF  TELESCOPE. 

Set  up  a  2-mm.  scale  in  a  vertical  position,  with  its  divisions 
in  a  plane  perpendicular  to  the  axis  of  the  telescope,  and  at  a 
distance  of  200  cm.  from  its  object-glass.  Focus  the  telescope 
on  the  divisions,  by  sliding  the  draw-tube  in  which  the  eye- 
piece is  set.  It  is  possible  now  to  see  the  image  in  the  tel- 
escope with  one  eye,  at  the  same  time  that  the  scale  itself  is 
seen  with  the  other;  and  by  slightly  turning  the  telescope,  if 
necessary,  to  see  the  divisions  of  the  image  apparently  super- 
posed on  those  of  the  scale. 

(a.)  When  this  can  be  easily  seen,  count  the  number  of 
large  divisions  and  fractions  which  are  covered  by  one  large 
division  of  the  image.  This  is  the  magnifying  power  of  the 
telescope  when  used  on  objects  at  this  distance. 

(b.)  Repeat  for  distances  of  150,  IOO,  75,  60,  50,  40,  and  30 
cm.  For  the  shorter  distances,  use  the  i-mm.  divisions  at  one 
end  of  the  scale. 

(<:.)  Plot  the  results  on  cross-section  paper,  using  distances 
as  abscissae  and  corresponding  magnifying  powers  as  ordinates. 
Draw  a  curve  through  the  points  thus  obtained,  which  will 
give  the  magnifying  power  at  any  distance  within  its  range. 

B.    MASS   OF   GIVEN   VOLUME   OF   AIR. 

To  determine  the  mass  of  a  volume  of  air  inclosed  in  a 
flask, over  water. 

(a.)  Mark  the  water-level  in  the  neck  of  the  flask  by  means 
of  a  mark  on  a  gummed  label,  and  measure  its  height  (h) 
above  the  water-level  in  the  beaker. 

(7>.)  Read  the  barometer  (see  Exp.  2,  a\  From  its  height 
(H)  and  (h),  find  the  pressure  on  the  inclosed  air  in  cm.  of 
mercury. 


8  MASS   OF   GIVEN   VOLUME   OF   AIR.  [B 

(r.)  The  inclosed  air  may  be  assumed  to  be  saturated  with 
aqueous  vapor.  Find  from  No.  15  of  Whiting's  tables  the 
maximum  pressure  of  aqueous  vapor  for  the  working  temper- 
ature. Subtracting  this  from  (b)  gives  the  pressure  due  to 
the  dry  air  in  the  flask. 

(d.)  Empty  the  flask,  wipe  off  any  moisture  on  the  outside, 
and  weigh  on  trip  scales.  Fill  with  water  to  the  mark  on  the 
neck,  and  weigh  again.  Find  from  table  22  the  volume  of  I 
gram  of  water  at  the  working  temperature,  and  calculate  the 
volume  of  the  water,  which  is  that  of  the  air  previously 
inclosed. 

Why  should  the  interior  of  the  flask  not  be  dried  before 
weighing  it  empty? 

(e.)  Using  the  formula  for  (Wd)  in  (c),  Exp.  2,  with  the 
proper  pressure  substituted  for  H-f,  calculate  the  mass  of  the 
inclosed  volume  of  air. 


l]  USE    OF    BALANCE.  9 


i.   USE   OF   BALANCE. 

Determination  of  the  weight  in  air  of  a  billiard-ball  (cellu- 
loid or  ivory). 

Put  the  ring  in  which  the  ball  is  to  be  set  to  keep  it  from 
rolling  about,  in  the  left  pan,  and  its  counterpoise  in  the  right 
pan. 

(a.)  Raise  the  beam  of  the  balance  by  means  of  the  lever 
in  front.  Move  the  hand  near  one  of  the  pans  if  necessary, 
until  the  pointer  swings  over  a  range  of  something  like  ten 
scale  divisions.  Then  close  the  case,  read  and  record  5,  7,  or 
any  odd  number  of  turning-points  of  the  pointer,  in  terms  of 
the  divisions  as  numbered  on  the  scale,  estimating  tenths  of 
the  small  divisions  as  closely  as  possible.  Unless  successive 
turning-points  differ  by  a  fairly  constant  amount,  showing  that 
the  vibration  is  regularly  decreasing,  the  series  should  be 
repeated. 

The  second  observer  may  read  turning-points  at  the  same 
time,  from  the  rear  of  the  case,  by  setting  up  a  plane  mirror 
parallel  to  and  facing  the  scale.  Use  both  eyes,  and  look  on 
both  sides  of  the  central  post  at  once,  in  order  to  avoid 
parallax. 

Average  the  turning-points  to  the  right,  and  those  to  the 
left,  separately.  The  average  of  these  two  results  is  the 
resting-point  of  the  pointier. 

(&)  Put  the  ball  in  the  ring  on  the  left  pan,  and  add  weights 
to  thfc  right  pan  until,  on  raising  the  beam,  the  pointer  swings 
about  a  point  near  the  middle  of  the  scale.  Determine  the 
resting-point. 

Always  handle  the  weights  with  pincers,  taking  care  not  to 
bruise  or  scratch  them.  Never  change  the  weights  on  the 
pan  without  first  lowering  the  beam. 

(c^)  Increase  or  decrease  the  weights  used  by  two  or  three 


to  CALIBRATION    OF    BAKODEIK.  [2 

centigrams,  so  as  to  move  the  resting-point  toward  the  center 
of  the  scale,  then  determine  this  new  resting-point.  Calculate 
the  change  in  weight,  in  milligrams,  required  to  change  the 
resting-point  by  one  scale-division.  This  is  called  the  sensi- 
tiveness of  the  balance  for  the  load  used. 

(d.)  Calculate  the  true  weight  of  the  ball  —  namely  the 
weights  which  would  give  the  resting-point  found  in  (a). 
Carry  out  the  result  to  milligrams,  and  correct  it  by  applying 
to  the  indicated  values  of  the  weights  used,  the  corrections 
given*  on  the  box. 

(e.)  The  result  given  above  is  the  weight  required,  if  the 
balance-arms  are  of  equal  length.  To  test  this,  weigh  the  ball 
in  the  right  pan.  Then  (G.  and  S.,  p.  100)  calling  the  balance 
arms  R  and  L  and  the  weights  of  the  ball  in  the  corresponding 
pans  Wv  and  W^ 


=  V'W1/W, 

and  the  true  weight  of  the  ball,  W  =  Wa(R/L). 

In  future  weighings  with  this  balance,  the  ratio  of  the  arms 
as  found  above  should  be  used  if  it  differs  from  unity  by  an 
amount  great  enough  to  affect  the  last  figure  desired  in  the 
result. 

2.   CALIBRATION  OF   BARODEIK.     PART  i. 

The  Barodeik  is  an  ordinary  balance,  having  a  hermetically- 
sealed  flask  suspended  from  one  scale-pan,  and  from  the  other, 
as  a  counterpoise,  a  glass  plate  so  chosen  as  to  have  a  surface 
about  equal  to  the  exterior  surface  of  the  flask.  The  reading 
of  the  balance  pointer  on  a  properly  graduated  scale  gives  the 
density  of  the  surrounding  air. 

To  find  the  difference  between  the  barodeik  reading  and 
the  true  density  of  the  air. 

(a.)  Set  and  read  the  barometer  (G.  and  S.,  p.  155).  The 
complete  setting  and  reading  should  be  repeated,  by  each 


2]  CALIBRATION    OF    BARODEIK.  1  I 

observer  independently,  until  the  readings  agree  reasonably 
well,  and  their  average  taken. 

(b.)  Read  the  wet  and  dry  bulb  thermometers.  Apply  cor- 
rections to  the  readings  as  indicated,  and  calculate  dew-point 
by  the  formula  given  on  the  instrument.  Then  from  No.  15 
of  Whiting's  tables  find  the  pressure  of  aqueous  vapor  in  the 
air,  remembering  that  "dew-point"  means  the  temperature  at 
which  the  aqueous  vapor  now  in  the  air  would  saturate  it;  or 
the  existing  pressure  of  aqueous  vapor  in  the  air  would  be  the 
maximum  pressure  if  the  temperature  were  lowered  to  the 
dew-point. 

(c.)  From  (a)  and  (&}  calculate  the  density  of  the  air.  The 
mass  of  one  cu.  cm.  of  dry  air,  at  o°  (C)  and  76  cm.  pressure, 
is  0.001293  grams.  The  mass  of  the  same  volume  of  aqueous 
vapor,  under  the  same  conditions,  is  f  as  much.  Then  if  h  be 
the  barometer  height,/  the  pressure  of  aqueous  vapor,  and  t 
the  temperature,  the  mass  of  dry  air  in  I  cu.  cm.  of  moist  air 
is: 

TT_r 

W,  =  0.001293  -  —  £• 
3  I  +at     76 

(a  is  the  coefficient  of  expansion  of  a  gas.) 
And  the  mass  of  aqueous  vapor  in  the  same  volume  is: 

Wv  =  (I)o.oo,  293  -L 


The  sum  of  these  two  is  the  required  density.  (Deschanel, 
p.  400-401.) 

(df)  Read  the  barodeik. 

Do  not  touch  the  instrument,  but  by  moving  the  hand  near 
one  of  the  scale-pans,  set  up  a  small  vibration.  Then  close 
the  case,  and  determine  the  resting-point  of  the  pointer,  which 
is  the  density  of  the  air  as  indicated  by  the  instrument. 

(/.)  Record  the  difference  between  the  reading  thus  ob- 
tained and  the  true  density  from  (c),  with  the  proper  sign,  so 


12  CALIBRATION   OF    BARODEIK.  [3 

that  when  added  algebraically  to  the  observed  reading  it  will 
give  the  true  density  of  the  air. 

This  is  the  absolute  correction  for  the  scale-division  to  which 
it  applies. 

3.    CALIBRATION   OF   BARODEIK.     PART    2. 

Relative  calibration  of  barodeik  scale. 

(a.)  Read  the  instrument  as  in  Part  I  (d).  Repeat  with  a 
2  eg.  weight  on  the  right-hand  scale-pan,  then  with  the  same 
weight  on  the  left  scale-pan. 

(b.)  Repeat  with  a  5  eg.  weight. 

Finish  the  observations  before  beginning  the  following  cal- 
culations, so  as  to  make  room  for  those  working  on  Part  I. 

(c.)  Using  the  exterior  volumes  of  flask  and  plate,  as  given 
on  the  instrument,  calculate  the  changes  in  the  density  of  the 
air  which  would  produce  the  same  effects  on  the  instrument  as 
the  putting  of  the  separate  weights  on  the  right  pan,  and  on 
the  left  pan. 

From  these  results  construct  a  table  of  corrections,  with  the 
proper  signs,  for  the  different  resting-points  observed. 

Note  that  this  is  a  relative  calibration.  That  is,  it  gives  the 
corrections  to  be  applied  to  certain  readings,  as  compared 
with  one  reading  (namely,  that  when  no  weights  were  used), 
which  is  assumed  correct. 

(d.)  In  Part  I,  the  absolute  correction  for  a  certain  reading 
was  found.  That  reading  was  not  far  from  the  one  assumed 
correct  above,  so  the  same  absolute  correction  may  be  applied 
to  the  latter.  By  means  of  this,  convert  the  table  of  relative 
corrections  (c)  into  a  table  of  absolute  corrections.  This  com- 
pletes the  absolute  calibration  of  the  instrument. 

(e.)  Plot  on  cross-section  paper  the  readings  on  the  barodeik 
scale  as  abscissae,  and  the  relative  corrections  of  (c)  as  ordinates, 
but  on  a  much  larger  scale.  Draw  a  curve  through  the 


5]  HYDROSTATIC    BALANCE.  13 

plotted  points,  which  will  give  the  correction  for  any  reading 
on  the  scale. 

Show  how  the  curve  can  be  made  to  indicate  absolute  cor- 
rections, instead  of  relative,  by  moving  the  horizontal  axis  of 
reference  up  or  down  by  a  proper  amount.  This  converts  it 
into  an  absolute  calibration  curve  for  the  instrument. 


4.  WRITTEN    EXERCISE. 

In  place  of  the  regular  laboratory  exercise,  write  out  the 
following  requirements  in  the  note-book,  and  hand  in  copy 
as  usual. 

Make  the  statements  clear  and  complete,  and  illustrate  by 
diagrams  when  possible.  Ask  for  assistance  on  any  points 
which  are  not  understood  after  a  reasonable  amount  of  study. 

1.  Why  is  an  odd  number  of  turning-points  used  in  deter- 
mining the  resting-point  of  a  balance  pointer? 

2.  Suggest  any  consideration  which  would  cause  the  sen- 
sitiveness of  a  balance  to  change  for  a  change  of  load. 

3.  Show   that    the    calculation    of   the    weight   in    i(d)    is 
simply  a  process  of  interpolation. 

4.  Show    that   the    method    used    for  finding   the   resting- 
point  of  a  balance  pointer   gives  a  better   result  than  could 
be    obtained    by    reading   the   position    of   the   pointer   after 
allowing  it  to  come  to  rest. 

5.  Give  a  full   explanation  of  the  action  of  the  barodeik. 
Why  should  the  flask  and  plate  have  equal  surfaces  exposed 
to  the  air? 

5.  HYDROSTATIC   BALANCE. 

Determination  of  mass,  volume,  and  density  of  billiard 
ball.  Use  the  same  balance,  weights,  and  ball  as  in  (i). 
The  resting-point  with  pans  unloaded  should  be  determined 


14  THERMOMETER    CALIBRATION.  [6 

anew  each  time  the  balance  is  used.  The  sensitiveness  need 
not  be  re-determined  unless  the  load  is  quite  different. 

Put  the  arch  over  the  left  pan,  and  on  it  a  beaker  of  water. 
Suspend  the  ball  in  its  wire  cage  from  the  hook  above  the 
pan  so  that  the  cage  is  entirely  immersed,  and  only  the  fine 
suspending  wire  cuts  the  surface  of  the  water. 

Take  the  temperature  of  the  water  before  and  after  the 
weighing,  and  consider  the  mean  of  the  two  to  be  the  tem- 
perature of  both  water  and  ball  during  the  experiment. 

(a.)  Weigh  as  in  (i),  and  read  the  barodeik. 

(b.)  Remove  the  ball,  fill  up  the  beaker  until  the  suspend- 
ing wire  of  the  cage  is  immersed  to  the  same  point  as  before, 
and  weigh  the  cage. 

(c.)  Calculate  the  volume  of  the  ball,  assuming  volume  of 
I  gram  water  =  one  cu.  cm.  as  an  approximate  value  to  be 
used  in  the  subsequent  work. 

(d.}  Calculate  the  following: 

Weight  of  air  displaced  by  the  ball  when  weighed  in  (i). 

Weight  of  air  displaced  by  the  weights  used,  assuming 
their  density  to  be  8.4. 

Weight  of  the  ball  in  vacuo. 

Mass  of  the  ball. 

(e.~)  Calculate  the  weight  of  air  displaced  by  the  weights 
representing  the  weight  of  the  ball  in  water  in  (a)  and  (^). 
From  this  find  what  the  ball  would  have  weighed  in  water, 
if  the  weighing  had  been  done  in  vacuo. 

(f.)  Find  the  weight  in  vacuo  of  the  water  displaced  by 
the  ball,  and  from  table  23,  calculate  the  exact  volume  of  the 
ball.  From  this  volume  and  the  mass,  calculate  the  density 
of  the  ball  at  the  temperature  of  the  determination. 

6.  THERMOMETER  CALIBRATION— ABSOLUTE. 

Calibration  of  a  100°  thermometer,  reading  to  i°,  which 
is  to  be  qsed  in  subsequent  work. 


6]  THERMOMETER    CALIBRATION.  1 5 

(a.)  Set  a  wire-gauze  cone  into  the  mouth  of  a  beaker^ 
and  fill  it  with  crushed  ice.  Put  the  thermometer  bulb  into 
the  middle  of  the  ice,  and  let  it  remain  for  a  few  minutes 
after  the  reading  has  become  apparently  constant. 

Record  the  reading  (to  o°.  i)  and  the  difference  between 
it  and  o°  with  the  proper  sign  as  the  correction  to  the 
reading  at  o°. 

To  avoid  parallax  in  the  readings,  always  turn  the  stem 
so  that  the  reflected  images  of  the  divisions  in  the  mercury- 
thread  can  be  seen,  apparently  as  prolongations  of  the  divi- 
sions themselves. 

(b.)  See  that  the  steam-heater  contains  a  sufficient  supply 
of  water.  Hang  the  thermometer  in  the  central  vertical  tube, 
with  the  IOO°  mark  just  above  the  cork,  arranging  the  tubes 
so  that  the  steam  will  pass  up  the  central  tube,  down  the 
space  between  the  tubes,  and  out  at  the  bottom  of  the  outer 
tube.  Trjus  the  thermometer  stem  is  surrounded  by  steam, 
and  this  again  by  a  jacket  of  steam. 

After  steam  has  issued  for  a  few  minutes,  adjust  the  lamp 
so  as  to  give  a  very  feeble  current  Why  is  this  necessary? 

Record  the  reading  as  before,  when  it  has  become  steady. 
Read  the  barometer,  and  find  corresponding  temperature  of 
steam  from  table.  Record  the  difference  between  this  and 
the  observed  reading,  with  the  proper  sign,  as  the  correction 
at  100°. 

(c.)  Let  the  thermometer  cool  slowly  to  about  the  tem- 
perature of  the  room,  then  repeat  (a).  If  the  result  differs 
from  that  previously  obtained,  ask  for  advice. 

(</.)  Ask  for  assistance  in  breaking  off"  a  thread  of  mercury 
about  50°  in  length. 

1.  Set  the  lower  end  at  the  o°  mark  and  read  the  upper  end. 
Set  the  upper  end  at  the  100°  mark  and  read  the  lower  end. 

2.  Repeat  with  a  thread  slightly  longer  or  shorter,  accord- 
ing a.s  the  first  was  too  short  or  too  long, 


1 6  THERMOMETER  CALIBRATION. 

3.  Calculate  for  each  case  what  the  readings  would  have 
been  if  the  ends  of  the  thread  had  been  set  at  the  true  o° 
and   1 00°  marks. 

4.  The  mean  of  the  four  readings  in  (3)  gives  the  middle 
point  of  the  thermometer,  or  the  true  50°  point.     From  this 
find  the  correction  at  50°. 

(*?.)  Break  off  a  thread  25°  in  length,  and  using  the  o°  and 
50°  marks  as  starting-points,  find  as  in  (d)  the  first  quarter- 
point,  or  true  25°  point,  and  the  corresponding  correction. 

(/)  With  the  50°  and  100°  marks  as  starting-points,  find 
the  third  quarter-point,  or  true  75°  point,  and  the  correction. 

(g.}  Plot  on  cross-section  paper  the  divisions  on  the  ther- 
mometer as  abscissae,  and  the  corrections  on  a  much  larger 
scale  as  ordinates.  Draw  as  smooth  a  curve  as  possible 
through  the  plotted  points.  This  gives  a  calibration-curve 
for  the  instrument. 

7.  THERMOMETER  CALIBRATION— COMPARATIVE. 

Calibration  of  a  50°  thermometer  reading  to  o°.i,  by  com- 
parison with  a  standard.  This  standard  is  a  thermometer 
like  the  one  to  be  calibrated,  but  which  has  been  carefully 
compared  with  a  reliable  standard  It  should  always  be 
returned  to  the  desk  after  using. 

Tie  your  thermometer  to  the  standard  with  soft  cotton 
twine,  winding  it  between  the  stems  so  as  to  separate  them 
slightly.  Put  the  bulbs  nearly  opposite  each  other;  and  see 
that  corresponding  divisions  are  as  nearly  opposite  as  is  con- 
sistent with  this  condition.  Suspend  the  two  securely  with 
the  bulbs  in  the  middle  of  a  kettle  of  water,  and  steady  the 
stems  by  catching  them  loosely,  without  pressure,  in  a  clamp. 
The  thermometers  are  to  be  read  by  a  paper-tube  telescope, 
which  slides  easily  on  the  vertical  rod  of  an  iron  stand.  This 
should  be  set  with  its  object-glass  at  a  distance  of  40-50  cm. 


7]  CALORIMETRY.  17 

from  the  thermometers,  which  should  be  perpendicular  to  its 
axis.  When  taking  a  reading,  always  set  the  telescope  so 
that  the  top  of  the  mercury  column  appears  in  the  middle 
of  the  field  of  view,  not  near  its  upper  or  lower  edge,  in 
order  to  avoid  parallax. 

(a.}  Take  a  careful  series  of  readings  to  o°.oi,  at  intervals 
of  2°  or  3°,  from  about  12°  to  45°.  Keep  the  water  well 
stirred,  and  keep  the  temperature  fairly  constant  for  a  few 
minutes  before  each  reading.  A  good  plan  is  to  take  a 
preliminary  reading  of  each  thermometer,  in  order  to  see 
about  where  the  reading  is  going  to  come.  The  two  exact 
readings  can  then  be  made  so  quickly  as  to  be  practically 
simultaneous. 

Read  again  in  a  few  seconds,  taking  the  thermometers  in 
reverse  order.  Repeat  if  necessary  until  the  differences 
obtained  from  two  such  readings  agree  fairly  well. 

($.)  Let  the  observers  change  places,  and  take  a  similar 
descending  series,  cooling  the  water  by  dipping  out  hot  and 
adding  cold. 

(c.}  Ask  to  see  the  calibration  curve  of  the  standard  used, 
and  from  it  construct  a  table  of  corrections  to  your  thermome- 
ter at  the  temperatures  used. 

(d.}  Plot  a  calibration  curve  as  in  (5). 

CALORIMETRY— GENERAL. 

The  calorimeter  apparatus  used  consists  of  a  light  copper 
cup,  the  calorimeter  cup  proper,  and  a  brass  water-jacket, 
which  is  wrapped  with  felt  and  closed  with  a  wooden  cover. 
The  cup  rests  on  three  corks  in  the  central  space  of  the 
jacket,  and  is  thus  surrounded  by  a  jacket  of  air,  except  for 
the  cork  supports  and  the  wooden  cover,  both  poor  con- 
ductors. A  central  opening  in  the  cover,  above  the  cup,  is 
closed  by  a  cork  through  which  the  thermometer  passes. 


1 8  WATER    EQUIVALENT    OF    CALORIMETER.  L8 

Two  stirrers,  one  for  the  cup  and  one  for  the  jacket,  are 
attached  to  a  common  rod  which  is  moved  vertically  by  means 
of  a  lever  attached  to  a  wooden  stand  on  which  the  apparatus 
is  placed. 

Use  the  thermometer  which  was  calibrated  in  (7),  and  read 
it  with  a  telescope  to  o°.oi,  as  was  done  there.  This  will  be 
called  the  "first  thermometer."  A  second  thermometer  will 
be  furnished  when  necessary,  to  take  the  temperature  of  a 
substance  which  is  to  be  put  into  the  calorimeter.  When  this 
is  done,  the  second  thermometer  should  be  compared  with  the 
first  in  a  water-bath  at  nearly  the  same  temperature,  in 
order  to  find  the  correction  to  be  applied  to  its  reading. 
Substances  to  be  poured  into  the  calorimeter  should  be  within 
i°  or  2°  of  the  temperature  o'f  the  room,  in  order  to  guard 
against  change  of  temperature  in  pouring. 

Before  putting  either  thermometer  into  a  hot  liquid,  always 
test  with  a  100°  thermometer  to  make  sure  that  its  tempera- 
ture is  under  50°. 

The  difference  in  temperature  between  cup  and  jacket  is 
usually  made  as  small  as  possible  by  adjusting  the  temperature 
of  the  water  in  the  jacket  to  that  necessary  for  the  cup.  The 
rate  of  change  of  temperature  of  the  cup,  due  to  radiation 
across  the  air-space  which  is  caused  by  this  difference,  is  thus 
made  as  small  as  possible,  and  being  fairly  constant,  can  be 
observed  and  allowed  for. 

All  weighings  should  be  made  accurate  to  about  0.2  per 
cent,  which  can  be  done  on  trip  scales  except  where  otherwise 
specified. 

8.   WATER   EQUIVALENT   OF   CALORIMETER. 

Determination  of  the  thermal  capacity  of  the  cup,  with  its 
stirrer  and  the  portion  of  the  thermometer  which  is  immersed 
in  water  when  the  cup  is  nearly  filled,  as  in  ordinary  use. 


8]  WATER    EQUIVALENT    OF    CALORIMETER.  19 

This  being  the  number  of  grams  of  water  to  which  the  appa- 
ratus is  thermally  equivalent,  is  called  its  "water  equivalent." 

(a.)  Weigh  a  beaker  containing  enough  water  to  half  fill 
the  cup,  and  set  it  on  the  stand  beside  the  calorimeter  so  that 
the  second  thermometer  can  be  hung  with  its  bulb  in  the 
water.  Adjust  the  thermometers  so  that  either  can  be  read 
easily  by  merely  turning  the  telescope,  without  altering  its 
focus. 

Weigh  the  cup,  fill  ^  full  of  water  at  about  40°,  and  weigh 
again.  Fill  the  jacket  with  water  at  the  same  or  a  somewhat 
higher  temperature,  pouring  it  in  through  a  small  funnel.  Be 
careful  always  to  avoid  getting  water  into  the  air-space  in  the 
jacket,  and  also  to  avoid  wetting  the  felt  wrapping. 

(£.)  While  one  observer  keeps  the  stirrers  in  gentle  but 
continuous  motion  (one  stroke  in  4  or  5  seconds  is  usually 
sufficient)  let  the  other  read  the  thermometer,  at  intervals  of 
one  minute,  for  4  or  5  minutes.  By  this  time  the  difference 
between  successive  readings  should  be  quite  constant.  At 
some  time  during  the  series,  stir  the  water  in  the  beaker  with 
the  thermometer  bulb,  and  read  its  temperature  between  two 
readings  of  the  first  thermometer.  Now  raise  the  cork  from 
the  calorimeter  cover,  and  pour  in  enough  water  from  the 
beaker  to  nearly  fill  the  cup.  This  should  be  done  by  the  one 
who  has  been  handling  the  stirrer,  and  so  timed  that  the  mixture 
is  made  at  a  half  minute,  between  two  readings.  Resume  the 
stirring  at  once,  and  continue  the  thermometer  readings  at 
each  minute  for  a  few  minutes  as  before. 

(c.^  From  the  average  rate  of  change  of  temperature  of  the 
cup  before  and  after  making  the  mixture,  calculate  the  tem- 
perature of  the  cup  at  the  moment  before  the  mixture  was 
made,  and  at  the  moment  after,  assuming  that  the  change  in 
temperature  due  to  the  mixture  was  instantaneous. 

Reweigh  the  beaker  to  find  weight  of  cold  water  used. 

Denoting  by  x,  the  thermal  capacity  of  the  apparatus, 


2O  WATER    EQUIVALENT    OF    CALORIMETER.  [8 

which  is  the  quantity  to  be  determined,  construct  an  equation 
involving  the  following  quantities: — 

1.  Quantity  of  heat  lost  by  the  hot  water  in  the  cup. 

2.  Quantity  of  heat  lost  by  cup,  stirrer,  and  thermometer. 

3.  Quantity  of  heat  gained  by  cold  water  poured  in. 
From  this  equation  find  x.     Use  corrected  temperatures  in 

calculating. 

(e,)  Repeat  with  initial  temperature  a  few  degrees  lower, 
and  take  the  mean  of  the  two  results. 

(/.)  As  a  check  on  the  preceding  result,  calculate  the 
thermal  capacity  of  the  cup  and  stirrer  from  their  weights  and 
the  specific  heats  of  copper  and  brass,  which  may  be  found  in 
the  tables.  To  this  must  be  added  the  thermal  capacity  of 
the  immersed  portion  of  the  thermometer,  which  may  be 
experimentally  determined  with  sufficient  accuracy  as  follows: 

Set  up  the  calorimeter,  leaving  out  the  cork,  with  water  at 
the  room  temperature  in  both  cup  and  jacket,  having  first 
weighed  the  cup  and  contained  water.  Record  the  tempera- 
ture when  it  has  become  steady,  then  take  out  the  thermom- 
eter and  immerse  it,  to  the  same  depth  as  is  usual  in  the  cup, 
in  water  at  about  45°.  After  a  few  minutes,  read  the  temper- 
ature to  o°.i,  then  as  quickly  as  possible  take  out  the  ther- 
mometer and  put  it  into  tke  usual  position  in  the  calorimeter, 
shaking  off  superfluous  water  on  the  way.  Stir  until  the 
temperature  becomes  steady  and  record  the  reading.  Then 
calculate  the  thermal  capacity  of  the  thermometer  according 
to  the  general  method  used  in  (d). 

(£•.)  If  this  result  differs  from  the  final  result  in  (e)  by  more 
than  one  unit,  ask  for  advice.  If  it  does  not,  use  the  mean  of 
the  two  as  the  water  equivalent  of  the  apparatus  in  future 
work. 

The  different  sets  of  apparatus  are  nearly  enough  of  the 
same  dimensions  so  that  this  result  may  be  used  with  any 
of  them. 


I01  HEAT   OF  SOLUTION.      AMMONIUM    NITRATE.  21 

9.    SPECIFIC    HEAT   OF   LEAD    SHOT. 

Set  up  the  apparatus,  take  observations  and  calculate 
results  as  in  Exp.  8,  (a),  (/;),  (c)  and  (if),  with  the  following 
modifications:  — 

(#.)  For  the  beaker  of  water,  substitute  a  copper  cup  con- 
taining about  700  grams  of  dry  shot  Fill  the  calorimeter 
cup  only  about  %  with  water,  to  avoid  risk  of  running  it  over 
when  the  shot  is  poured  in. 

(&)  When  the  shot  is  poured  in,  the  temperature  will  not 
be  well  equalized  by  the  ordinary  stirring,  as  the  stirrer  will 
not  penetrate  the  mass  of  shot.  It  is  necessary  to  unhook 
the  thermometer  from  its  support  and  stir  the  shot  very  care- 
fully with  its  bulb  for  about  one  minute.  Then  replace  the 
cork,  hang  up  the  thermometer  and  resume  the  readings,  one 
of  which  has  thus  been  omitted.  From  this  point  the  ordinary 
stirring  is  sufficient. 

(c.)  In  writing  the  equation,  x  should  now  denote  the 
specific  heat  of  the  shot,  and  the  thermal  capacity  of  the 
apparatus  is  known. 

(d.}  Repeat  with  a  fresh  portion  of  shot,  and  a  higher  initial 
temperature  of  about  46°. 

Wash  the  shot  well  after  use,  and  spread  it  in  a  thin  layer 
on  a  cloth  to  dry. 

10.    HEAT   OF   SOLUTION.     AMMONIUM    NITRATE. 

The  quantity  of  heat  absorbed  in  the  solution  of  one  gram 
of  a  substance  is  called  its  heat  of  solution.  If  heat  is  given 
out  in  the  solution,  the  quantity  is  considered  negative. 

If  the  temperature  of  the  salt  after  solution  be  different 
from  that  at  which  it  was  poured  into  the  water,  it  will  be 
necessary  to  consider  its  specific  heat  also.  According  to  the 
following  method  the  heat  of  solution  and  the  specific  heat 


22  LATENT    HEAT   OF   VAPORIZATION.  [10 

are  both  determined,  although  the  former  is  the  main  object 
of  the  experiment. 

On  one  of  the  Becker  balances  weigh  out  two  portions  of 
the  salt,  each  of  10  grams,  to  o.oi  gram.  This  can  be  con- 
veniently done  by  putting  two  equal  pieces  of  paper  in  the 
two  pans,  and  pouring  the  salt  on  one  of  them.  Never  put 
the  salt  directly  into  the  pan.  Put  each  portion,  after  weigh- 
ing, into  a  smallest-size  beaker  which  is  thoroughly  dry. 

This  amount  of  salt,  when  dissolved  in  about  2OO  grams  of 
water,  will  lower  its  temperature  a  little  over  3°.  It  is  best 
to  have  the  cup  about  3°  degrees  warmer  than  the  jacket, 
because  the  larger  part  of  the  salt  dissolves  in  a  few  seconds, 
so  that  the  cooling  during  this  time  is  small;  and  the  tem- 
perature being  then  reduced  to  about  that  of  the  jacket,  there 
is  no  cooling  during  the  longer  time  required  for  the  complete 
solution  of  the  salt. 

(a.)  With  the  jacket  at  the  room  temperature  and  the  cup 
about  3°  higher,  proceed  as  in  (8).  In  removing  the  second 
thermometer  from  the  beaker  of  salt,  after  taking  its  tempera- 
ture, see  that  no  salt  is  removed.  After  pouring  in  the  salt, 
stir  rather  vigorously  to  hasten  the  solution,  and  record  the 
temperature  when  it  has  become  steady. 

(£.)  Make  a  similar  trial  with  the  second  portion  of  salt, 
having  the  jacket  at  40°  or  higher.  Make  sure  that  there  is 
the  proper  difference  of  temperature  between  the  cup  and 
the  jacket  before  beginning  the  observations. 

(c.)  Calling  the  specific  heat  of  the  salt  x  and  its  heat  of 
solution  yt  write  the  proper  equation  for  each  case,  and  solve 
the  two  equations  for  x  and  y. 

CAUTION. — Do  not  leave  the  solution  standing  in  the  copper 
cup.  Wash  it  out  as  soon  as  possible. 

ii.    LATENT   HEAT   OF   VAPORIZATION. 
Examine  the  construction  of  the  steam-trap,  and  see  that 


II] 


LATENT    HEAT   OF   VAPORIZATION. 


when  held  vertically,  with  the  exit  tube  down,  it  will  catch 
and  discharge  through  the  side  vent  all  water  which  is  con- 
densed on  the  way  from  the  boiler,  and  permit  only  dry  steam 
to  issue  from  the  exit.  Fill  the  boiler  nearly  full  of  water, 
heat  it  to  boiling,  then  remove  the  lamp  until  ready  to  use 
the  steam. 

The  steam  will  be  condensed  in  a  copper  tube,  closed  at  the 
bottom,  which  projects  downward  from  the  lower  surface  of 
the  cork  into  the  calorimeter  cup.  This  tube,  closed  with  a  dry 
cork  and  well  dried  on  the  outside,  is  to  be  weighed,  before  and 
after  using,  on  one  of  the  delicate  balances.  These  balances 
must  be  handled  with  especial  care.  See  that  the  pans  are 
protected  by  sheets  of  filter-paper.  Adjust  the  weights  to  the 
nearest  centigram,  then  use  the  rider  to  obtain  the  nearest 
milligram.  Never  touch  weights  or  rider  without  first  arrest- 
ing the  pans. 

The  cut  is  a  section  through  the  cork, 
showing  the  proper  arrangement  of  the 
trap  and  tube  when  in  operation.  See 
cut.  The  cup  must  be  filled  until  the 
water  rises  almost,  but  not  quite,  to  the 
lower  surface  of  the  cork,  so  that  the 
copper  tube  may  be  immersed  as  deeply 
as  possible.  The  proper  amount  of 
water  should  be  determined  by  a  prelim- 
inary trial. 

Lay  a  small  piece  of  blotting-paper  over  the  hole  in  the 
corkj  and  punch  a  corresponding  hole  through  the  paper, 
using  the  exit  tube  as  a  punch.  When  steam  is  issuing  from 
the  tube,  an  occasional  drop  of  water  forms  at  the  mouth, 
which  will  be  absorbed  by  the  paper.  The  object  of  these 
precautions  is  to  make  sure  that  the  water  which  is  collected 
and  weighed  in  the  tube  is  simply  that  which,  in  condensing 
on  the  walls  of  the  tube,  gives  up  its  heat  of  vaporization  to 
the  water  in  the  cup. 


i 


24  HEAT   OF   CHEMICAL   COMBINATION.  12 

Put  in  the  cup  the  proper  amount  of  water  at  a  temperature 
about  3°  below  that  of  the  jacket,  which  should  be  at  the 
room  temperature.  Set  the  lamp  under  the  boiler,  and  adjust 
its  height  so  that  when  steam  is  issuing  freely  the  water  stands 
at  about  25  cm.  in  the  manometer  tube.  This  gives  a  vigorous 
current  of  steam,  without  raising  the  boiling  point  too  much. 
The  stirring'of  the  jacket  may  be  dispensed  with,  and  the  cup 
stirred  directly  by  hand.  After  reading  temperatures  for  a 
few  minutes,  put  the  steam-trap  in  position,  with  the  side 
vent  pointing  away  from  the  thermometer,  and  press  down  so 
tightly  that  no  steam  escapes  around  the  cork.  Stir  gently 
until  the  temperature  has  risen  about  6°,  then  remove  the 
trap,  continue  stirring,  and  record  the  highest  temperature. 
Corrections  for  cooling  are  practically  unnecessary,  since  the 
range  of  temperature  is  distributed  equally  above  and  below 
that  of  the  jacket. 

In  calculating  results,  remember  to  account  for  the  quantity 
of  heat  given  out  by  the  condensed  steam  in  cooling  from  the 
boiling-point  (which  may  be  taken  as  100°)  to  the  temperature 
of  the  cup. 

Make  a  second  trial,  which  should  give  a  good  result,  with 
the  practice  in  handling  the  apparatus  acquired  during  the 
first  trial. 

12.    HEAT   OF   CHEMICAL   COMBINATION. 

Determination  of  the  heat  generated  by  the  combination  of 
NaOH  with  HC1  to  form  NaCl. 

A  0.5  normal  solution  of  each  of  the  above  compounds  is 
furnished.  By  a  normal  solution  is  meant  one  which,  in  1,000 
grams  of  the  solution,  contains  a  weight  of  the  element  which 
is  to  enter  into  the  new  combination  equal  in  grams  to  its  atomic 
weight.  Thus  the  normal  solution  of  NaOH  is  a  solution 
which  contains,  in  1,000  grams,  40  grams  (23+16+1)  of 
NaOH,  or  23  grams  of  Na.  The  0.5  normal  solutions  used 
contain  one-half  of  these  proportions. 


J3]  COEFFICIENT   OF   EXPANSION    OF   A   LIQUID.  25 

It  is  evident  that  if  equal  weights  of  these  solutions  be 
mixed,  the  reaction  will  be  just  completed,  and  the  result  will 
be  a  neutral  solution  of  NaCl.  The  solutions  are  to  be  mixed 
in  the  calorimeter  cup  at  as  nearly  as  possible  the  same 
temperature,  and  the  resulting  rise  of  temperature  noted. 
The  alkali  should  be  placed  in  the  cup,  and  the  acid  added  to 
it.  The  acid  being  immediately  neutralized,  will  then  have  no 
action  on  the  metal  of  the  cup. 

Weigh  out  100  grams  of  the  NaOH  solution  in  the  cup, 
and  the  same  amount  of  the  HC1  solution  in  the  beaker.  The 
latter  should  be  weighed  out  roughly  at  first,  poured  back 
into  the  bottle,  then  the  wet  beaker  counterpoised  and  the 
amount  weighed  accurately.  This  will  then  be  quite  accu- 
rately the  weight  which  is  afterward  poured  into  the  calori- 
meter. A  small  error  is  introduced  by  taking  the  second 
thermometer  out  of  the  beaker  after  reading  its  temperature, 
but  this  may  be  neglected. 

If  care  has  been  taken  not  to  handle  the  cup  and  beaker 
any  more  than  necessary,  the  two  temperatures  should  be 
very  nearly  the  same  when  ready  for  use.  Since  the  amounts 
used  are  equal,  it  may  be  safely  assumed  that  the  resulting 
solution  of  NaCl  has  risen  to  the  final  temperature  from  the 
mean  of  the  two  initial  temperatures. 

A  direct  determination  of  the  specific  heat  of  the  NaCl 
solution  is  impracticable.  The  value  0.987,  which  has  been 
-calculated  by  interpolation  from  tabulated  results,  may  be 
used  for  this  case. 

Mfcke  two  trials,  and  calculate  for  each  the  number  of  heat 
units  which  would  have  been  evolved  had  normal  solutions 
been  used. 

13.    COEFFICIENT   OF   EXPANSION   OF  A   LIQUID. 

The  method  consists  in  determining  the  weights  of  alcohol 
filling  a  specific  gravity  bottle  at  several  different  temperatures, 


26  COEFFICIENT   OF   EXPANSION   OF   A   UQUID.  [*3 

and  from  these  data  calculating  the  coefficient  of  expansion. 
Four  determinations  should  be  made,  at  intervals  of  about  8°, 
beginning  with  the  room  temperature. 

(a.)  Fill  the  bottle  with  alcohol  and  set  it  on  a  platform  in 
a  kettle  of  water,  so  that  the  water  comes  as  high  as  the 
alcohol  in  the  bottle.  See  that  the  stopper  is  set  in  very  lightly. 
Hang  your  50°  calibrated  thermometer  in  the  bath  beside 
the  bottle,  and  keep  the  bath  well  stirred  for  about  5  minutes. 
The  temperature  of  the  bath  should  remain  constant  within 
o°.i  during  this  time,  and  at  the  end  of  it  the  alcohol  will 
have  the  same  temperature  within  o°.i. 

Set  the  stopper  in  tightly,  using  no  more  pressure  than 
necessary,  take  the  bottle  out  of  the  bath,  wipe  dry  and  weigh 
to  o.ooi  gram  on  one  of  the  delicate  balances. 

(b.)  Repeat  with  the  bath  at  about  each  of  the  higher  tem- 
peratures selected,  holding  the  temperature  steady  for  10 
minutes  by  holding  the  lamp  under  the  kettle  for  a  few 
seconds  occasionally.  Before  putting  the  bottle  in  the  bath, 
loosen  the  stopper  to  allow  the  excess  of  alcohol  to  escape, 
and  try  it  occasionally  to  see  that  it  remains  loose  during  the 
warming.  Careful  trial  has  shown  that,  after  this  treatment, 
the  temperature  of  the  alcohol  at  the  center  of  the  bottle,  as 
shown  by  the  thermometer  in  the  stopper,  is  about  o.°  I  lower 
than  that  of  the  bath,  and  therefore  the  average  temperature 
of  the  alcohol  is  the  same  as  that  of  the  bath  to  within  o°.i. 

(c.)  Empty  the  alcohol  into  the  bottle  from  which  it  was 
taken,  dry  the  bottle  with  the  jet-pump,  and  weigh.  (Ask  for 
directions  as  to  the  use  of  the  jet-pump.) 

(*/.)  Find  the  weight  of  alcohol  filling  the  bottle  at  each 
temperature.  Plot  the  results,  with  the  corrected  tempera- 
tures, starting  from  o°,  as  abscissae,  and  changes  in  weights 
as  ordinates.  Assuming  that  the  expansion  is  uniform,  draw 
the  straight  line  which  best  represents  the  plotted  points,  and 
from  it  find  the  weight  filling  the  bottle  at  o°. 


14]  VOLUMENOMETER.  2/ 

(e.)  From  the  weights  filling  the  bottle  at  o°  and  at  40°, 
calculate  the  relative  volume  of  a  given  weight  of  alcohol  at 
40°,  referred  to  its  volume  at  o°.  From  this  calculate  the 
coefficient  of  cubical  expansion  of  the  alcohol  for  the  given 
range  of  temperature.  (Coefficient  of  expansion  is  expansion 
of  I  cu.  cm.  from  o°  to  i°.) 

(/.)  Note  that  what  you  have  obtained  is  not  the  absolute 
coefficient  for  alcohol,  since  the  bottle  also  expands.  Find 
from  the  tables  the  coefficient  of  cubical  expansion  of  glass, 
and  by  applying  it  to  the  above  result,  find  the  absolute 
coefficient  for  the  alcohol.  Remember  that  the  expansion  of 
the  interior  of  a  glass  bottle  is  equal  to  the  expansion  of  the 
same  volume  of  solid  glass. 

14.   VOLUMENOMETER. 

Density  of  NaCI  Crystals. 

The  apparatus  is  designed  to  measure  the  volume  Fof'air 
inclosed  above  a  certain  mark  on  the  glass  tube.  The  given 
volume  is  inclosed  over  mercury,  and  its  pressure  p  meas- 
ured by  the  manometer.  By  raising  or  lowering  the  manom- 
eter tube,  the  volume  of  the  inclosed  air  is  changed  by  an 
amount  vlt  which  can  be  measured,  and  the  corresponding 
pressure  pl  is  read.  By  Boyle's  law,  pv  —  const.  Hence 
pV=p^  Vlt  where  Vl=  F-H'i,  from  which  V,  the  original 
volume,  is  determined.  The  temperature  is  assumed  to  re- 
mam  constant. 

CAUTION. — Put  no  dirty  mercury  into  the  instrument,  and 
see  that  there  is  a  mercury-tight  tray  on  the  floor  under- 
neath it. 

(a.)  Having  the  brass  tube  unscrewed  from  the  upper  end 
of  the  fixed  glass  tube,  so  that  the  latter  is  open  to  the  air, 
set  the  movable  tube,  pouring  in  more  mercury  if  necessary, 


28  VOLUMENOMETER.  [14 

so  that  the  top  of  the  column  in  the  fixed  tube  is  at  the 
middle  mark.  See  that  the  brass  tube  is  empty  except  for  a 
plug  of  cotton.  Clean  off  the  old  grease  from  its  end,  and 
from  the  shoulder  on  the  brass  cap,  smear  a  little  fresh  grease 
evenly  on  the  screw-threads  of  the  cap,  and  screw  on  the 
tube,  making  sure  that  it  fits  tightly  against  the  shoulder. 

Readjust  the  open  tube  so  that  the  edge  of  the  meniscus 
in  the  closed  tube  is  again  at  the  mark.  The  volume  now 
inclosed  above  the  mercury  is  the  volume  Fto  be  determined. 

Read  the  position  of  the  top  of  each  column  on  the  meter 
rod  by  means  of  a  straight  edge  laid  across  its  face,  and  from 
the  difference  in  level  and  the  barometer  height,  find  the 
pressure/  on  the  inclosed  air. 

(b)  Raise  the  open  tube,  pouring  in  more  mercury  if  neces- 
sary, until  the  mercury  rises  to  the  upper  mark  in  the  closed 
tube.  Let  stand  a  few  minutes  to  allow  the  inclosed  air  to 
return  to  the  temperature  of  the  surrounding  air,  then  read 
both  columns.  Let  stand  five  minutes,  and  read  again.  Any 
difference  is  probably  due  to  leakage  in  the  screw-joint,  which 
must  be  made  tight. 

Repeat  the  setting  three  or  four  times,  bringing  the  column 
in  the  closed  tube  to  the  mark  and  reading  the  open  tube  each 
time. 

From  the  heights  of  the  two  columns,  find  the  pressure  pv 

(c.)  Lower  the  open  tube,  catching  the  overflow  in  a  beaker, 
until  the  column  in  the  closed  tube  sinks  to  the  lower  mark. 
Read  both  columns  as  before,  and  find  the  pressure  p,2  for 
this  case. 

(d)  Bring  the  inclosed  air  back  to  nearly  atmospheric 
pressure,  and  unscrew  the  brass  tube.  Take  out  the' cotton 
plug  from  the  brass  tube,  fill  it  nearly  full  of  clean  NaCl 
crystals,  and  replace  the  plug  so  that  they  will  not  fall  out. 
Repeat  (a),  (b),  and  (c),  the  volume  V  now  being  less  than 
before  by  the  volume  of  the  crystals. 


15]  CALIBRATION    OF    THE    TUBE.  2Q 

(e.)  A  separate  tube  is  furnished  which  contains  the  volumes 
z\  and  2A2  between  marks  corresponding  to  those  on  the  vol- 
umenometer. To  find  these  volumes,  fill  the  tube  to  the 
marks  with  mercury,  from  a  separate  supply,  pour  into  a 
weighed  beaker  and  weigh,  using  Becker  balance.  The  vol- 
ume of  one  gram  of  mercury  at  the  room  temperature  may 
be  found  in  the  tables. 

(/)  Calculate  the  value  of  Ffor  each  case,  and  so  find  the 
volume  of  the  crystals  used. 

Weigh  the  crystals,  and  calculate  their  density. 

15,  16,  17,  18.  MEASUREMENTS  OF  CHANGE  OF 
VOLUME  WITH  TEMPERATURE. 

The  apparatus  used  is  practically  a  thermometer  with  a 
large  bulb,  open  at  the  end.  This  opening  can  be  closed  by 
a  stopper  consisting  of  a  flat  surface  of  rubber  which  is  pressed 
against  it  by  springs.  The  constants  of  the  instrument  are 
determined  in  the  first  two  experiments,  for  use  in  the  two 
succeeding. 

The  instrument  furnished  to  each  pair  of  students  will  be 
used  by  them  only,  until  the  four  experiments  are  completed. 

15.   CALIBRATION   OF   THE   TUBE. 

Always  handle  mercury  over  a  mercury-tight  tray. 

(a.)  The  cm.  divisions  nearest  the  ends  of  the  scale  will  be 
considered  as  the  ends  of  the  scale  to  be  used,  corresponding 
to  the  o°  and  100°  marks  on  the  ordinary  thermometer  scale. 
Pour  a  little  mercury  into  the  bulb  and  run  it  into  the  tube, 
so  as  to  secure  a  continuous  thread  as  long  as  the  scale. 
Adjust  by  pouring  out  a  drop  at  a  time  from  the  end  of  the 
tube.  Read  the  length  of  the  column,  find  its  weight,  to  I 
eg.,  and  from  this  find  the  volume  of  the  tube  between  the 
two  ends  of  the  scale. 


3O  EXPANSION    OF    THE   THERMOMETER.  [16 

(b.)  Find  the  middle  and  quarter  points  by  the  method  used 
in  6,  and  also  find  the  eighth  points,  midway  between  the 
quarter  points.  The  bore  of  the  tube  is  apt  to  vary  consider- 
ably in  different  parts,  so  that  the  quarter  points  will  not  give 
a  sufficiently  good  calibration. 

(<:.)  On  a  sheet  of  co-ordinate  paper,  lay  off  the  divisions 
of  the  scale  as  abscissae,  and  at  each  of  the  points  on  the 
scale  found  above,  including  the  upper  end  of  the  scale,  erect 
an  ordinate  which  will  represent  the  volume  of  the  tube  from 
the  zero  up  to  that  point.  A  curve  through  these  points  will 
then  give  the  volume  of  the  tube  between  any  two  points  on 
the  scale.  Such  a  curve  can  be  best  drawn  with  a  flexible 
ruler,  such  as  a  strip  of  spring  brass  on  edge,  held  at  the  ends 
so  as  to  pass  through  all  the  points. 

16.    EXPANSION   OF   THE   THERMOMETER. 

(a.)  Fill  the  thermometer  with  clean  mercury,  getting  the 
right  quantity  by  trial  so  that  the  top  of  the  column  will  rise 
in  the  tube  to  the  lower  part  of  the  scale  when  the  stopper  is 
set  in  place  so  as  to  inclose  no  air-bubbles. 

Always  steady  the  bulb  with  one  hand  when  handling  the 
stopper,  as  a  slight  twist  of  the  bulb  may  break  the  tube. 

(b.)  Immerse  the  bulb  in  a  water-bath  so  that  the  tube  will 
be  vertical,  and  suspend  your  100°  calibrated  thermometer 
beside  it.  Read  the  height  of  the  column  in  the  tube  at 
intervals  of  about  10°  up  to  60°,  holding  the  temperature  of 
the  bath  steady  at  each  chosen  point  for  a  few  minutes  after 
the  reading  of  the  expansion  thermometer  has  become  steady, 
and  reading  the  temperature  to  o°.i.  Then  cool  the  bath,  and 
take  a  similar  set  of  readings  at  temperatures  intermediate 
between  those  previously  used. 

If  there  is  time,  heat  the  bath  again,  taking  a  second  series 
of  readings. 


17]  EXPANSION    OF    COTTON-SEED    OIL.  3! 

Find  the  weight  of  mercury  used,  by  carefully  pouring  it 
out  into  a  weighed  beaker,  and  from  this  find  its  volume. 
Use  trip  scales. 

(c.)  Plot  the  results,  using  corrected  temperatures  starting 
from  o°  as  abscissae,  and  corresponding  volumes  of  the  tube, 
from  the  volume-curve,  as  ordinates.  Draw  the  straight  line 
which  best  represents  the  plotted  points.  Through  the  point 
where  this  line  intersects  the  ordinate  at  o°,  draw  an  axis 
parallel  to  the  axis  of  abscissae.  The  apparent  expansion  of 
the  mercury  in  the  thermometer,  from  o°  to  any  given  temper- 
ature, will  then  be  represented  by  the  corresponding  ordinate 
measured  from  this  axis  to  the  plotted  line. 

(d.)  At  the  highest  temperature  used,  erect  an  ordinate 
from  this  axis  which  will  represent  the  absolute  expansion  of 
the  mercury  used,  from  o°  to  this  temperature,  and  diaw  a 
line  from  this  point  to  the  o°  point.  The  difference  between 
corresponding  ordinates  to  the  two  lines  gives  the  expansion 
of  the  apparatus.  Construct  a  third  plot  which  will  represent 
this  expansion  measured  from  a  horizontal  axis. 

(>.)  Assuming  that  the  apparatus  is  made  entirely  of  glass, 
calculate  the  coefficient  of  expansion  of  the  glass. 

17.    EXPANSION   OF   COTTON-SEED   OIL. 

Fill  the  instrument  with  oil,  in  the  same  way  that  it  was 
filled  with  mercury  in  the  last  experiment.  There  will  be  less 
trouble  with  air-bubbles  if  the  oil  is  poured  into  the  bulb 
slowly,  in  a  steady  stream. 

(a.)  Set  the  thermometer  in  a  water-bath  as  before,  and  take 
a  similar  set  of  ascending  and  descending  readings.  The  oil  is 
such  a  poor  conductor  that  it  will  be  necessary  to  keep  the 
temperature  of  the  bath  constant  at  each  reading  for  a  much 
longer  time  than  with  the  mercury. 

($.)  Plot  the  observations  as  in  16,  (<r).     To  correct  for  the 


32  EXPANSION    OF    Nad    CRYSTALS.  .  L1^ 

expansion  of  the  apparatus,  measure  from  this  line  at  some 
convenient  temperature  an  ordinate,  in  the  proper  direction,  to 
represent  the  expansion  of  the  thermometer  from  o°  to  that 
temperature,  taken  from  the  final  plot  in  16.  Then  a  line 
drawn  through  this  point  and  the  point  where  the  first  line 
intersects  the  ordinate  of  o°  will  represent  the  true  expansion 
of  the  oil. 

(c.)  Calculate  the  coefficient  of  expansion  of  the  oil.  The 
volume  at  o°  of  the  oil  used  can  easily  be  calculated,  but  if 
the  result  be  found  in  terms  of  the  volume  at  the  lowest  tem- 
perature used,  it  will  be  nearly  enough  the  same. 

Leave  the  oil  in  the  thermometer  for  the  next  experiment. 

'  '18,   EXPANSION  OF   NaCl   CRYSTALS. 

The  bulb  is  now  to  be  filled  with  a  weighed  amount  of 
NaCl  crystals,  leaving  the  spaces  between  them  filled  with  the 
oil  used  in  the  last  experiment. 

Set  the  thermometer,  still  containing  the  oil,  in  the  bath 
with  the  bulb  partly  immersed,  its  axis  being  vertical,  and 
heat  to  70°. 

Pick  out  and  weigh  enough  clean  NaCl  crystals  to  more 
than  fill  the  tube.  Remove  the  stopper  from  the  bulb  and 
drop  in  the  crystals  one  at  a  time.  The  heat,  and  gentle 
stirring  around  with  a  bit  of  wire,  will  set  free  the  air-bubbles 
which  at  first  are  caught  by  the  rough  surfaces.  After  getting 
in  the  proper  amount  of  crystals,  fill  up  with  oil  arid  take  a 
set  of  readings  as  in  17.  Weigh  the  remaining  crystals,  and 
calculate  the  volume  used  from  the  density  found  in  14. 

Plot  the  observations,  and  correct  the  line  obtained  for  the 
expansion  of  the  thermometer,  and  of  the  volume  of  oil  used. 

From  the  resulting  line  representing  the  true  expansion  of 
the  crystals,  calculate  their  coefficient  of  expansion. 


i8]  ELECTRICITY.  33 


ELECTRICITY—GENERAL   DIRECTIONS. 

To  set  up  a  Daniell  cell :  The  zinc  plate  is  left  standing  in 
water  in  the  outer  cup,  and  the  inner  porous  cup,  filled  with 
CuSO4  solution  and  containing  the  copper  plate,  is  kept  soak- 
ing in  a  jar  of  the  solution.  Clean  off  any  scale  from  the 
surface  of  the  zinc  by  rubbing  it  with  a  rag  or  the  fingers, 
under  the  tap,  and  put  it  in  place.  Rinse  off  the  outside  of  a 
porous  cup,  which  is  filled  to  within  3  or  4  cm.  of  the  top  with 
CuSO4  solution,  and  put  it  inside  the  zinc.  Fill  the  space 
around  it  with  dilute  H2SO4  from  one  of  the  labeled  bottles. 

When  you  have  finished  using  the  cell,  put  it  away  as  it 
was  found,  returning  the  acid  to  the  bottle. 

If  gas  is  given  off  when  the  acid  is  first  poured  in,  the  zinc 
needs  amalgamating.  Ask  for  directions. 

When  these  precautions  are  observed,  the  electromotive 
force  and  resistance  of  the  cell  will  be  very  nearly  the  same 
at  one  time  as  at  another.  The  resistance  may  change  slightly 
for  a  few  minutes  on  first  setting  up. 

In  setting  up  a  circuit  of  any  kind,  clean  all  the  connections, 
including  those  at  the  battery  terminals,  with  knife,  sandpaper, 
or  file.  Never  make  connections  by  twisting  the  ends  of  wires 
together,  as  the  resistance  of  such  a  connection  is  very  variable. 
Always  use  binding  screws,  and  make  sure  that  the  ends  of 
the  wire  are  firmly  held  by  the  screws. 

If  at  any  time  the  current  seems  to  be  variable,  or  too 
weak,  examine  the  connections  first,  as  they  are  very  likely  to 
be  the  seat  of  the  trouble. 

Do  not  move  tangent  galvanometers  without  first  asking 
permission;  and  handle  them  carefully,  to  avoid  breaking  the 
silk  fiber  suspensions.  If  the  pointer  does  not  swing  freely 
after  adjusting  the  leveling  screws,  ask  for  assistance. 

In   using  galvanometers   with   pivot   suspension,   tap  the 


34  TANGENT   GALVANOMETER.  [19-21 

instrument  with  some  soft  object  before  taking  a  reading. 
Raise  the  needle  from  the  pivot  before  leaving  the  instrument. 

Binding  screws  and  connecting  wires  will  be  found  at  the 
northwest  corner  of  the  room;  return  them  to  their  places 
after  use. 

In  making  connections,  do  not  use  wires  which  are  longer 
than  necessary,  and  make  a  practice  of  arranging  wires  so  that 
currents  in  opposite  directions  shall  be  as  near  each  other  as 
possible,  in  order  to  minimize  effect  on  galvanometer  needle. 
Use  wires  of  small  diameter,  except  in  special  cases  where  it 
is  desirable  to  make  the  resistance  of  connections  as  small  as 
possible. 

In  using  resistance  boxes,  take  care  to  protect  the  plugs, 
and  the  tapered  holes  into  which  they  fit,  from  bruises  and 
dirt.  Never  interchange  plugs  between  boxes  without  per- 
mission. Keep  boxes  covered  when  not  in  use. 

19,  20,  21.  TANGENT   GALVANOMETER. 

GENERAL  THEORY. — A  unit  magnetic  pole  is  defined  as  one 
which,  when  placed  in  air  at  a  distance  of  I  cm.  from  a  pole 
of  equal  strength,  will  exert  upon  it  a  force  of  I  dyne. 

A  unit  current  is  defined  as  one  which,  flowing  in  a  wire 
I  cm.  long,  bent  into  an  arc  of  I  cm.  radius,  will  act  on  a 
unit  magnetic  pole  at  the  center  of  the  arc  with  a  force  of 
I  dyne. 

We  shall  speak  of  the  field  at  the  center  of  the  coil,  due 
to  the  current  flowing  in  it,  as  the  force  in  dynes  exerted  by 
the  current  on  a  unit  pole  at  the  center  of  the  coil;  let  this 
be  denoted  by  F. 

For  a  unit  current,  flowing  in  a  circular  coil  of  I  turn, 
radius  I  cm.,  F=2n;  for  n  turns,  F=2*n.  If  the  radius  of 
the  coil  be  r,  F=2xn/r;  since  the  field  due  to  a  given  length 
of  wire  varies  as  i/r2,  but  the  length  of  wire  in  the  coil  is  r 


19]  TEST    OF    TANGENT    LAW.  35 

times  as  great  This  value  of  Ft  namely  the  field  produced 
at  the  center  of  the  coil  by  a  unit  current  flowing  in  the  coil, 
is  called  the  constant  K  of  the  galvanometer. 

//denotes  the  horizontal  component  of  the  earth's  magnetic 
field  at  the  center  of  the  coil.  Then  if  the  coil  be  set  with 
its  plane  vertical  and  in  the  magnetic  meridian,  the  two  fields 
of  force  denoted  by  F  and  H  will  act  in  directions  at  right 
angles  to  each  other,  upon  each  pole  of  the  needle.  Assum- 
ing the  needle  to  be  so  short  that  the  field  at  each  pole  is 
the  same  as  at  the  center  of  the  coil,  if  the  current  c  cause 
the  needle  to  deflect  through  an  angle  a  from  its  zero  posi- 
tion, show  that  in  general  tan  a  =  FIH=cKjH.  From  this 
c  =  (HjK)  tan  a. 

c  is  here  expressed  in  c.  g.  s.  units.  The  commercial 
unit,  the  ampere,  is  o.i  of  the  c.  g.  s.  unit.  Hence  if  c  be 
expressed  in  amperes,  c  =  (\oHjK)  tan  a.  \®H\K  is  called 
the  reduction  factor  R  of  the  galvanometer. 

19.  TEST   OF   TANGENT   LAW. 

(a.)  Set  the  galvanometer  with  its  plane  in  the  magnetic 
meridian.  With  the  instruments  used,  this  is  done  by  bring- 
ing the  ends  of  the  pointer  to  o°  and  180°.  Set  up  a  Daniell 
cell,  and  connect  it  in  series  with  the  galvanometer  and 
resistance  box.  On  the  galvanometer  set  the  plug  of  the 
small  switch-board  in  the  hole  marked  50;  the  current  then 
flows  through  a  coil  of  50  turns.  Take  out  plugs  from  the 
box  until  the  deflection  is  about  50°. 

Determine  the  deflection  by  reading  both  ends  of  the 
pointer,  to  o°.i,  then  reversing  the  terminals  to  get  a  deflec- 
tion in  the  opposite  direction  and  reading  both  ends  again. 
The  mean  of  the  four  readings  is  the  deflection  required. 
Increase  the  resistance  in  the  box  so  as  to  obtain  deflections 
at  intervals  of  5°  or  6°,  down  to  about  10°. 


36  REDUCTION    FACTOR.  [20 

In  what  follows,  the  electromotive  force  and  resistance  of 
the  cell  are  supposed  to  remain  constant  throughout  the 
experiment.  To  test  this,  repeat  the  first  reading.  If  it 
does  not  agree  with  the  former  value,  repeat  the  series  in 
the  same  order  as  before  to  a  point  where  the  new  values 
do  agree  with  the  old. 

(£.)  Show  the  truth  of  the  relation  expressed  on  the  pre- 
ceding page,  tan  a  —  F]H.  By  combining  this  with  Ohm's 
law,  E—cr,  where  r  is  the  total  resistance  in  the  circuit,  show 
that  EjR  =  r  tan  #  =  r/cotan  a. 

If  then  you  make  a  plot  with  resistances  as  abscissae  and 
cotangents  of  corresponding  angles  of  deflection  as  ordinates, 
the  points  obtained  should  lie  on  a  straight  line  if  the  gal- 
vanometer obeys  the  tangent  law. 

(<r.)  Make  such  a  plot  from  your  observations,  using  box 
resistances  as  abscissae,  and  show  how  to  obtain  from  it  the 
constant  resistance  in  the  circuit  outside  the  box.  Do  this 
first  by  reading  it  directly  from  the  plot,  and  then,  more 
accurately,  by  constructing  the  equation  of  the  line  and 
calculating  from  it. 

What  does  this  resistance  include? 

The  same  galvanometer  must  be  used  for  the  next  two 
experiments. 

20.  REDUCTION    FACTOR. 

(#.)  Set  up  your  galvanometer  with  its  needle  as  nearly  as 
possible  over  one  of  the  numbered  brass  nails  on  the  tables. 
Straighten  out  two  of  the  copper  wire  spirals  from  the 
voltameter  cells,  clean  and  smooth  them  well  with  fine 
emery  paper,  and  re- wind  on  a  brass  rod  provided  for  the 
purpose.  Do  not  touch  the  wire  with  the  fingers  except  at 
the  ends,  after  cleaning;  catch  one  end  through  the  small 
hole  in  the  brass  rod,  and  having  the  other  end  secured, 


20]  REDUCTION   FACTOR.  37 

stretch  the  wire  while  winding,  so  as  to  get  a  smooth  spiral. 
If  the  surface  of  the  wire  flakes  off  in  cleaning,  ask  for  new 
wire.  Clean  the  copper  plates  of  the  voltameters,  replace 
them  and  the  spirals,  and  fill  the  cells  with  fresh  electrolyte 
solution  from  the  bottle. 

Connect  the  two  voltameters  in  series  with  a  Daniell  cell 
and  the  5 -turn  coil  of  the  galvanometer,  in  such  a  way  that 
copper  will  be  deposited  by  the  current  on  the  spirals. 
Insert  German  silver  wire  if  necessary  to  reduce  the  deflec- 
tion to  about  25°. 

(b.)  When  the  deflection  remains  practically  constant  for 
several  minutes,  lift  the  spirals  out  of  the  solution  and  see 
if  they  are  uniformly  covered  with  a  bright  and  clean  deposit. 
If  so,  plunge  them  at  once  in  a  beaker  of  clean  water,  then 
wash  thoroughly  under  the  tap  and  dry  by  gentle  heat,  no 
greater  than  may  be  easily  borne  by  the  hand. 

Weigh  the  spirals  separately  to  I  mg.  'on  the  delicate 
balance. 

The  battery  should  be  allowed  to  send  a  current  through 
some  convenient  resistance  while  not  in  use. 

(c.)  Replace  the  spirals  in  the  voltameters,  put  them  in 
circuit  as  before,  and  let  the  current  run  for  50  minutes  50 
seconds.  Read  the  deflection  at  each  minute,  reversing  its 
direction  at  about  the  middle  of  the  period.  Repeat  the 
washing,  drying,  and  weighing.  The  gain  in  weight  should 
be  the  same  for  the  two  spirals.  If  it  is  nearly  the  same, 
take  the  mean. 

-  (d.)  Remembering  that  the  weight  of  copper  in  grams 
deposited  by  the  current  in  the  given  time  is  numerically 
equal  to  the  current  strength  in  amperes,  calculate  the  reduc- 
tion factor  for  the  coil  used.  If  the  deflection  has  not 
remained  constant,  the  mean  of  the  tangents  of  the  observed 
angles  should  be  used  in  the  calculation.  However,  if  the 
variation  has  not  been  large,  the  tangent  of  the  mean  angle 
may  be  used  instead.  % 


38  CONSTANT.  [2! 

Do  not  replace  the  used  electrolyte  solution  in  the  bottle, 
but  pour  it  in  the  jar  containing  porous  cups. 

21.    CONSTANT. 

(a.)  To  calculate  the  constants  of  the  i-turn,  5-turn  and 
5O-turn  coils,  from  their  measured  dimensions. 

Measure  with  a  beam-compass  the  diameter  of  the  ledge  on 
the  face  of  the  ring,  which  is  the  same  as  the  diameter  of  the 
bottom  of  the  groove,  or  the  inner  diameter  of  the  coil.  Also 
measure  the  outer  diameter  of  the  wire  coil.  In  each  case 
take  several  measurements  along  different  diameters. 

The  first  layer,  in  the  bottom  of  the  groove,  consists  of  six 
turns  of  No.  14  wire,  so  connected  that  the  current  flows 
through,  all  six  in  parallel,  thus  making  one  effective  turn. 
This  is  the  i-turn  coil.  The  next  layer,  of  No.  16  wire,  con- 
sists of  two  wires  in  parallel  wound  four  times  around,  thus 
making  four  turns.  These  eight  wires  just  fill  the  width  of 
the  groove,  as  do  the  six  wires  of  the  first  layer.  The  four 
turns  and  the  one  may  be  connected  in  series,  making  the 
5-turn  coil.  Outside  of  this  are  45  turns  of  No.  22  wire,  which 
may  be  connected  in  series  with  the  five  turns,  to  make  the 
5O-turn  coil. 

The  coils  of  galvanometers  I  and  2  are  not  wound  accord- 
ing to  the  plan  here  described.  If  they  are  used,  ask  for 
the  plan. 

Samples  of  No.  14  and  No.  16  wire  are  furnished.  From 
the  measured  diameters  of  these,  and  the  inner  and  outer 
diameters  of  the  whole  coil,  find  the  mean  diameter  of  each 
coil.  Calculate  the  constant  for  each  coil. 

From  the  definition  of  the  constant  it  is  evident  that  the 
constant  for  two  coils  in  series  is  the  sum  of  the  two  con- 
stants taken  separately.  Find  thus  the  required  constants  for 
the  i-turn,  5-turn  and  5O-turn  coils. 


22]  VARIOMETER.  39 

(b.)  From  the  constants,  and  the  reduction  factor  for  the 
5-turn  coil,  calculate  the  reduction  factors  for  the  l-turn  and 
5O-turn  coils.  From  the  plot  of  Exp.  19,  take  a  pair  of  values 
of  r  and  cotan  a,  and  find  the  electromotive  force  of  the  cell 
used  from  these  and  the  proper  reduction  factor. 

(c.)  Find  the  value  of  H  at  the  point  where  the  needle  of 
the  galvanometer  was  situated  in  Exp.  20.  Locate  this  point 
by  measuring,  by  means  of  two  meter  rods  placed  end  to  end 
in  succession,  the  distance  (x)  from  the  west  wall  of  the 
laboratory,  (y)  from  the  south  wall,  and  (z)  from  the  floor. 
This  gives  the  three  co-ordinates  of  the  point,  referred  to 
rectangular  axes  with  origin  at  the  southwest  corner  of  the 
floor. 

22.    VARIOMETER. 

From  the  value  of  H  found  in  the  last  experiment,  to  deter- 
mine its  value  at  another  point  in  the  room.  The  point  to 
be  used  is  No.  10,  in  the  northeast  part  of  the  room.  The 
instrument  is  a  tangent  galvanometer,  adapted  to  the  purpose 
by  attaching  to  the  top  of  the  coil  a  carriage  rotating  in  a 
horizontal  plane  for  the  control  magnet.  As  the  strength  of 
this  magnet  is  supposed  to  remain  constant  throughout  the 
experiment,  it  must  not  be  jarred,  nor  brought  into  contact 
with  iron. 

(a.)  Release  the  galvanometer  needle  by  raising  the  pin  at 
one  side  of  the  scale,  and,  having  the  control  magnet  removed 
a 'short  distance  and  stood  up  vertically  to  neutralize  its 
magnetic  effect,  turn  the  coil  and  scale  until  the  pointer 
stands  nearly  at  o°.  Read  both  ends  accurately.  See  that 
there  are  no  masses  of  iron  on  the  table,  or  on  the  persons  of 
the  observers,  which  might  affect  the  readings.  Now  place 
the  control  magnet  in  its  carriage,  which  should  be  placed 
approximately  at  right  angles  to  the  plane  of  the  coil,  and 


4O  VARIOMETER.  [22 

clamped  just  firmly  enough  so  that  it  will  rotate  with  some 
difficulty. 

(b.)  Read  both  ends  of  the  pointer.  Lift  the  control  mag- 
net from  the  carriage,  and  after  reversing  its  direction,  replace 
it  without  moving  the  carriage.  Read  the  pointer  as  before. 
Calculate  the  true  deflection  in  each  direction  from  the  zero 
found  in  (a).  If  the  two  deflections  agree,  show  that  the 
axis  of  the  control  magnet  must  be  at  right  angles  to  the 
earth's  lines  of  force.  If  they  do  not  agree,  see  which  way 
to  rotate  the  carnage  in  order  to  make  the  difference  less,  and 
by  repeated  trials  reduce  it  to  a  small  quantity. 

How  could  this  method  be  used  to  set  the  plane  of  the  coil 
of  a  tangent  galvanometer  in  the  magnetic  meridian,  by  means 
of  the  current  in  its  coil  ? 

(<:.)  Let  each  observer  read  several  sets  of  deflections  in 
both  directions.  Take  the  mean  of  the  series  as  the  true 
deflection. 

(//.)  Leaving  the  carriage  clamped  in  position,  raise  the 
needle  from  the  pivot,  and  set  the  instrument  with  the  needle 
as  nearly  as  possible  at  the  point  where  the  value  of  H  is 
known.  Adjust  the  pointer  to  o°,  and  take  a  series  of  read- 
ings of  deflections  as  before.  If  a  galvanometer  is  in  use  near 
by,  determine  whether  it  will  affect  your  work  by  taking  a 
careful  reading  when  a  current  is  flowing  in  the  coil,  and 
again  when  it  is  not.  If  the  effect  is  appreciable,  try  to  obtain 
your  readings  when  no  current  is  flowing. 

(e.)  Again  set  the  instrument  in  its  first  position,  and  repeat 
the  first  set  of  readings.  If  the  result  does  not  agree  with 
that  previously  obtained,  it  is  evidence  that  the  strength  of 
the  control  magnet  has  probably  changed.  The  mean  of  the 
two  results  should  be  taken  as  corresponding  most  nearly  to 
the  condition  of  things  at  the  time  when  the  readings  in  (c) 
were  taken. 

(/.)  Noting  the  similarity  between  the  field  produced  at  the 


23]  VOLTMETER   CALIBRATION.  41 

center  of  the  coil  by  the  control  magnet  and  that  which  would 
be  produced  the-re  by  a  current  in  the  coil,  show  how  to  cal- 
culate the  ratio  of  the  values  of  H  at  the  two  points,  and  from 
this  its  absolute  value  at  the  new  point.  Find  the  co-ordinates 
of  this  point  as  before. 

23.    VOLTMETER   CALIBRATION. 

Any  galvanometer  may  be  used  as  a  voltmeter  if  its  resist- 
ance be  sufficiently  high.  A  mirror  galvanometer  is  here 
used,  its  deflections  being  measured  by  means  of  a  telescope 
and  scale. 

The  galvanometer  used  has  not  the  necessary  high  resist- 
ance. Its  coils  are  to  be  connected  in  series  with  a  resistance 
coil,  and  the  galvanometer  and  resistance  coil  taken  together 
then  constitute  the  voltmeter. 

The  method  used  for  the  calibration  depends  upon  the  fact 
that  if  a  constant  current  be  maintained  through  a  circuit  of 
given  resistance,  the  E.  M.  F.  between  any  two  points  on  that 
circuit  varies  as  the  resistance  included  between  them.  The 
voltmeter  is  so  arranged  as  to  read  the  E.  M.  F.  between 
various  pairs  of  points  on  such  a  circuit,  which  consists  of  a 
wire  of  uniform  resistance. 

(a.)  On  the  rheostat  provided,  each  of  the  six  loops  and 
the  stretched  wire  are  of  the  same  length,  and  may  for  the 
present  be  assumed  to  have  the  same  resistance.  This  equality 
will  be  tested  afterward.  Set  up  a  Daniell  cell  at  once,  and 
connect  it  to  the  terminals  A  and  B  of  the  rheostat,  in  order 
that  the  current  may  become  steady  by  the  time  the  readings 
are  to  be  made. 

Notice  that  if  all  three  of  the  plugs  provided  are  in  place, 
in  any  three  holes,  the  current  from  the  cell  will  flow  through 
four  meters  of  wire.  By  changing  the  plugs  from  one  end  of 
the  board  to  the  other,  the  straight  meter  on  which  the  sliding 


42  VOLTMETER   CALIBRATION.  [23 

contact  works  may  be  made  the  first,  second,  third  or  fourth 
of  the  four  meters;  so  that  the  sliding  contact  may  be  made 
to  move  from  one  end  to  the  other  of  the  four  meters  through 
which  the  current  is  flowing.  The  circuit  as  now  set  up  will 
be  called  the  main  circuit. 

(b.)  If  the  galvanometer  does  not  work  properly  at  any 
time,  do  not  attempt  to  adjust  it,  but  ask  for  assistance. 

Set  the  telescope  about  on  the  normal  to  the  mirror,  which 
may  be  easily  located  by  seeing  the  reflection  of  your  eye  in 
it.  Make  the  distance  of  the  scale  from  the  mirror  (measured 
to  the  axis  of  the  tube  containing  it)  150  cm.  Point  the 
telescope  tube  by  sighting  along  it,  directly  at  the  mirror. 
Raise  or  lower  the  scale  until  its  image  is  seen  in  the  mirror 
on  looking  along  the  tube.  If  this  is  carefully  done,  the 
image  of  the  scale  should  be  distinctly  seen  in  the  telescope 
when  it  is  properly  focused.  Move  the  telescope  sideways, 
making  the  last  small  adjustment  with  the  slow-motion  screw, 
until  the  image  of  the  center  division  of  the  scale,  marked  o 
or  20,  coincides  with  the  vertical  cross-wire  in  the  eye-piece. 

For  each  setting  the  true  deflection  is  to  be  obtained  by 
taking  the  mean  of  the  deflections  to  right  and  left  on  revers- 
ing the  current. 

(c.)  If  the  current  as  it  is  now  flowing  in  the  main  circuit 
causes  an  appreciable  deflection  of  the  galvanometer  needle 
(tested  by  making  and  breaking  the  circuit),  try  to  reduce  it 
by  proper  arrangement  of  connecting  wires. 

See  that  the  inner  edges  of  the  slides  to  which  the  galvano- 
meter coils  are  attached  are  at  division  10  on  the  scale,  if  the 
Edelmann  galvanometer  is  used;  if  the  other  instrument  is 
used,  set  the  outer  edges  of  the  slides  at  12.5. 

(d.}  It  will  be  best  now  to  make  a  sketch  on  paper  of  the 
whole  scheme  of  connections,  and  ask  to  have  this  verified 
before  setting  up.  For  this  purpose  the  four  meters  of  wire 
in  the  main  circuit  may  be  represented  as  a  single  straight 


23]  VOLTMETER   CALIBRATION.  43 

piece,  and  the  galvanometer  circuit  added  according  to  the 
following  description. 

Such  a  sketch  of  connections  should  accompany  the  notes 
of  each  of  the  following  experiments  on  electricity. 

Connect  the  resistance  coil  (500  ohms  for  the  Edelmann, 
1,000  for  the  other)  in  series  with  the  galvanometer  coils,  to 
make  up  the  necessary  high  resistance  for  the  galvanometer 
circuit  Connect  one  terminal  of  this  circuit  to  one,  terminal 
of  the  rheostat,  the  other  to  the  binding-screw  c,  at  the  end  of 
the  brass  rod  on  which  the  sliding  contact  moves.  The  gal- 
vanometer circuit  may  thus  be  put  in  parallel  with  any  desired 
portion  of  the  main  circuit  by  proper  arrangement  of  plugs. 

Include  in  the  galvanometer  circuit  a  mercury-cup  com- 
mutator, in  such  a  way  that  tipping  it  from  one  side  to  the 
other  will  reverse  the  direction  of  the  current  in  the  galvano- 
meter circuit,  without  affecting  that  in  the  main  circuit.  It 
must  evidently  be  put  between  the  rheostat  and  the  galvano- 
meter. Include  in  the  galvanometer  circuit  also  a  key  near 
the  telescope,  enabling  the  observer  to  make  and  break  that 
circuit  at  pleasure. 

It  is  evident  that  in  setting  up  these  connections,  small 
wires  may  be  used  throughout. 

(e.)  Reckoning  from  that  rheostat  terminal  to  which  the 
galvanometer  terminal  is  attached,  arrange  the  plugs  so  that 
the  stretched  meter  shall  be  the  first  of  the  four.  Let  one 
observer  set  the  sliding  contact  at  the  first  marked  point,  which 
is  20  cm.  from  the  end,  thus  putting  the  galvanometer  circuit 
in  parallel  with  20  cm.  of  the  main  circuit,  and  let  the  other 
read  the  deflection;  then  reverse  the  commutator  and  read 
the  opposite  deflection.  Repeat  for  the  other  marked  points, 
at  50  and  80  cm.  Learn  to  control  the  vibrations  of  the 
needle,  so  as  to  bring  it  to  rest  quickly,  by  proper  use  of  the 
key. 

Change  the  plugs  so  as  to  make  the  stretched  meter  sue- 


44  WHEATSTONE   BRIDGE.  [24 

cessively  the  second,  third  and  fourth  of  the  series,  and  for 
each  case  read  the  deflections  for  each  of  the  marked  points. 
Record  for  each  case  the  total  length  of  wire  on  the  main 
circuit  with  which  the  galvanometer  circuit  is  in  parallel,  and 
also  the  numbers  of  the  loops  used  in  making  up  each  com- 
bination for  the  main  circuit. 

The  observers  should  change  places  at  about  the  middle  of 
the  series. 

(/)  If  there  is  time,  repeat  the  series  in  reverse  order, 
moving  the  galvanometer  terminal  which  was  at  one  terminal 
of  the  rheostat  to  the  other  one.  The  lengths  of  wire  now 
included  between  the  galvanometer  terminals  will  be  measured 
from  the  opposite  end  of  the  four  meters.  Use  the  same 
combinations  as  before  in  making  up  the  main  circuit,  for 
convenience. 

If  the  resistance  in  the  galvanometer  circuit  remain  constant, 
show  that  the  currents  indicated  by  the  deflections  will  be 
proportional  to  the  E.  M.  F.'s  between  its  terminals;  and  that, 
if  the  resistance  be  sufficiently  large  compared  with  that  of 
the  main  circuit,  these  E.  M.  F.'s  will  be  proportional  to  the 
resistances  of  those  portions  of  the  main  circuit  included  be- 
tween the  terminals. 

Finally  connect  the  terminals  of  the  galvanometer  circuit 
directly  to  those  of  the  cell,  leaving  out  the  rheostat  entirely, 
and  read  the  deflection.  Show  that  the  current  here  indicated 
should  be  proportional  to  the  E.  M.  F,  of  the  cell,  as  deter- 
mined in  Exp.  21. 

Calculations  may  be  deferred  until  Exp.  25,  and  the  papers 
for  this  exercise  and  that  handed  in  together. 

24.     WHEATSTONE  BRIDGE. 

If  a  current  be  sent  through  two  wires  in  parallel,  there 
will  be  a  certain  E.  M.  F.  between  the  two  ends  a  and  b  of  this 


24]  WHEATSTONE    BRIDGE.  45 

portion  of  the  circuit.     If  a  point  c  be  chosen  on  one  of  the 
wires,  there  will  be  a  certain  E.  M.  F.  between  a  and  c.     A 
point  d  can  then  be  found  on  the  other  wire,  such  that  the 
c  same  E.  M.  F.  exists 

between  a  and  d.     It 
is  then  evident  that 
if  the  points  c  and  d 
d  be    connected    by   a 

wire,  there  will  be  no  E.  M.  F.  between  them  along  this  wire, 
and  hence  no  current  will  flow  in  the  wire.  The  point  d 
may  then  be  found  by  inserting  a  delicate  galvanoscope  as 
part  of  the  circuit  cd,  and  moving  d  along  adb  until  no  current 
is  indicated. 

By  applying  Ohm's  law  to  each  of  the  four  portions  ac%  cb 
ad>  db,  show  that  when  this  adjustment  has  been  made  rjx=. 
pjq,  where  r  denotes  the  resistance  of  ac,  x  that  of  cb,  etc. 

In  the  apparatus  used,  adb  is  a  uniform  stretched  wire  of 
German  silver  I  meter  long,  and  d  is  a  sliding  contact  on  this 
wire,  x  is  an  unknown  resistance  to  be  determined,  and  ris  a 
known  resistance  (for  this  experiment  a  coil  whose  resistance 
in  ohms  is  marked  upon  it). 

It  is  evident  that  a  constant  cell  is  not  required,  since  the 
relations  obtained  above  are  independent  of  the  current- 
strength  in  the  main  circuit  (They  will  not  be  true,  how- 
ever, for  rapid  variations  in  that  current,  since  Ohm's  law 
does  not  hold  for  such  cases.)  It  is  customary  to  use  a 
Leclanche  cell,  dry  cell,  or  other  inconstant  element,  and  to 
include  in  the  battery  circuit  a  key  which  is  only  depressed 
while  taking  an  observation,  since  such  cells  can  not  send  a 
strong  current  for  any  length  of  time  without  injury. 

Be  careful  never  to  connect  such  a  cell  so  that  it  will  send 
a  continuous  current. 

To  determine  the  resistance  per  cm,  of  No.  19  German 
silver  wire: — 


46  WHEATSTONE    BRIDGE.  [24 

(a.)  Set  up  the  apparatus  according  to  the  plan  outlined 
above,  with  about  two  meters  of  the  wire  as  the  unknown 
resistance.  Connect  the  ends  of  the  wire  to  the  proper  points 
on  the  circuit  through  binding-posts,  measuring  the  length  of 
wire  included  between  them.  In  connecting  the  wire  and 
box  to  the  bridge,  use  pieces  of  large  copper  wire  as  short  as 
possible,  and  well  cleaned  at  the  points  of  contact.  It  is  also 
well  to  have  the  resistances  of  the  connecting  wires  used  on 
the  two  sides  of  c  as  nearly  equal  as  possible  if  the  position  of 
d  is  near  the  middle  of  the  wire,  as  it  usually  is.  Small  wire 
should  of  course  be  used  for  the  galvanometer  and  battery 
circuits. 

Ask  to  have  your  scheme  of  connections  verified  before 
beginning  work. 

See  that  the  galvanometer  is  not  affected  by  a  current  in 
any  branch  of  the  circuit  except  its  own. 

(b.)  Find  by  trial  the  position  of  the  contact  on  the  bridge- 
wire  which  gives  zero  deflection  of  the  galvanometer.  In 
making  a  trial,  hold  the  sliding  contact  down  with  a  gentle 
pressure,  at  a  point  near  one  end  of  the  bridge,  then  depress 
the  battery  key  just  long  enough  to  note  the  direction  of  the 
deflection  produced.  Make  a  similar  trial  near  the  other  end 
of  the  bridge,  and  if  the  deflection  is  in  the  opposite  direction, 
the  position  sought  must  lie  between  these  limits. 

(c.)  Leaving  the  connecting  wires  in  position,  interchange 
the  wire  and  the  coil,  and  make  a  new  determination.  The 
sliding  contact  will  now  come  on  the  other  side  of  the  center 
of  the  bridge,  so  that  its  position  can  readily  be  found;  and 
the  mean  of  the  two  results  from  these  observations  will 
be  nearly  free  from  small  errors  due  to  lack  of  uniformity  in 
the  stretched  wire  and  scale,  and  to  the  resistance  of  the 
connections. 

(d)  Make  a  similar  pair  of  determinations  for  a  new  length 
of  wire,  shorter  by  a  few  cm.  than  the  first 


25]  TESTING   EQUALITY   OF  RESISTANCES.  47 

Calculate  from  each  result  the  resistance  of  the  wire  in 
ohms  per  cm.  Wire  cut  from  the  same  piece  will  be  used 
as  a  known  resistance  in  later  experiments,  for  the  determina. 
tion  of  small  resistances. 

(e.)  Measure  the  diameter  of  the  wire  with  the  micrometer 
caliper  at  several  points,  and  calculate  the  specific  resistance 
of  this  sample  of  German  silver — that  is,  the  resistance  of  a 
piece  I  sq.  cm.  in  cross-section  and  I  cm.  long.  This  value 
must  not  be  taken  as  applying  to  German  silver  in  general, 
since  different  samples  vary  greatly  in  specific  resistance. 

25.    TESTING   EQUALITY   OF   RESISTANCES. 

(a.)  Secure  the  rheostat  used  in  Exp.  23,  and  with  a  Wheat- 
stone  bridge  apparatus  similar  to  that  used  in  the  last  experi- 
ment, compare  the  resistance  of  each  of  the  six  loops  with 
that  of  the  stretched  meter.  The  stretched  meter  will  then  con- 
stitute the  r  branch  for  each  case,  and  the  loop  used,  the  x 
branch.  Connections  may  be  readily  made  by  means  of  the 
binding-screw  plugs  which  fit  the  blocks  of  the  rheostat.  See 
that  the  connecting  wires  are  properly  equalized,  and  use 
the  method  of  interchange  described  in  Exp.  24  (c). 

(b)  Assuming  that  the  wire  on  the  rheostat  is  of  uniform 
resistance  throughout,  calculate  the  length  of  each  loop,  that 
of  the  stretched  meter  being  100  cm. 

(c.)  For  each  of  the  combinations  of  loops  used  with  the 
voltmeter,  calculate  the  true  length  of  wire,  in  terms  of  the 
above  results,  and  the  true  position  of  the  slider  on  this  length 
for  each  reading.  Then  reduce  the  position  of  the  slider  to 
what  it  would  have  been  had  the  true  length  been  400  cm.,  by 
proportion.  A  correction  of  less  than  I  cm.  on  any  one  loop 
may  be  disregarded. 

Plot  the  results,  using  as  abscissae  the  corrected  lengths  of 
wire  in  the  main  circuit  with  which  the  voltmeter  was  in 


48  CALIBRATION    OF    RHEOSTAT.  [26 

parallel,  and  as  ordinates  the  corresponding  deflections  of  the 
voltmeter.  If  the  galvanometer  obeys  the  tangent  law,  show 
that  the  plot  should  be  very  nearly  a  straight  line,  so  that  any 
ordinate  will  be  proportional  to  the  corresponding  E.  M.  F. 

Show  how  to  convert  this  relative  calibration  curve  into 
an  absolute  one,  which  will  read  E.  M.  F.  in  volts,  by  means 
of  the  deflection  corresponding  to  the  known  E.  M.  F.  of  the 
Daniell  cell. 


26.    CALIBRATION   OF    RHEOSTAT. 

The  bridge  apparatus  here  used  is  a  more  compact  form, 
in  which  only  50  cm.  of  the  wire  is  stretched  along  a  scale, 
the  remainder  of  the  meter  being  made  up  of  two  portions  of 
25  cm.  each  which  may  be  connected  to  the  first  portion  by 
plugs.  This  connection  may  be  made  by  means  of  the  two 
plugs  in  three  different  ways,  so  that  the  50  cm.  along  the 
scale  may  be  either  end  or  the  middle  of  the  meter.  The 
latter  arrangement  will  be  most  used,  and  the  scale  is  num- 
bered to  correspond  to  it.  For  any  combination,  the  terminals 
of  the  meter  are  evidently  the  blocks  which  are  not  connected 
by  a  plug,  and  the  binding-screw  plugs  should  be  inserted  in 
these  blocks,  to  replace  the  binding-screws  at  the  ends  of  the 
wire  in  the  ordinary  form. 

A  rheostat  is  furnished,  the  coils  of  which  have  nominal 
resistances  of  I,  2,  2  and  5  ohms.  Determine  as  accurately 
as  possible  the  resistance  of  each  of  these  coils,  using  as  known 
resistances  the  corresponding  coils  on  a  standard  rheostat  of 
similar  pattern,  which  has  been  carefully  adjusted. 

After  finding  the  resistance  of  each  coil  separately,  deter- 
mine that  of  the  four  in  series  as  a  check. 

The  rheostat  which  has  thus  been  calibrated  will  be  used 
for  known  resistances  in  the  next  experiments. 


27]  RESISTANCE   OF   GALVANOMETER   COILS.  49 


27.    RESISTANCE   OF   GALVANOMETER   COILS. 

Spools  are  provided  whose  resistances  are  equal  respectively 
to  those  of  the  5o-turn  and  the  5-turn  coils  on  tangent  gal- 
vanometer No.  6.  Determine  accurately  the  resistance  of 
these  spools. 

The  resistance  of  the  5-turn  coil  is  so  small  that  the  ordinary 
method  can  not  be  relied  upon  to  give  good  results,  since  the 
resistance  of  the  connections  is  here  relatively  important. 
In  such  cases  a  substitution  method  may  be  employed  to 
advantage.  Set  up  the  bridge  with  the  spool  as  the  r  branch, 
and  a  piece  of  German  silver  wire  for  x,  of  a  length  which  will 
bring  the  slider  to  a  position  near  the  middle  of  the  bridge. 
Adjust  the  slider  accurately  for  balance.  Then  remove  the 
spool  and  substitute  for  it  a  piece  of  the  No.  19  wire,  whose 
resistance  per  cm.  is  known,  stretching  it  between  the  same 
binding-posts  which  held  the  spool,  so  that  the  connections 
are  exactly  as  before.  With  the  slider  in  the  same  place, 
adjust  the  length  of  wire  between  the  binding-posts  until 
balance  is  again  obtained.  It  is  then  evident  that  the  resist- 
ance of  the  length  of  wire  used  is  exactly  equal  to  that  of  the 
spool  for  which  it  was  substituted. 

This  method  should  be  used  always  for  low  resistances,  as 
well  as  for  any  work  where  great  accuracy  is  required,  since 
it  entirely  elimimites  errors  due  to  connections,  as  well  as  to 
faults  of  any  kind  in  the  bridge  itself. 

FQI-  the  5<D-turn  coil,  use  the  ordinary  bridge  method  with 
the  calibrated  rheostat  as  known  resistance. 

From  the  data  of  Exp.  19,  find  the  resistance  of  the  battery 
there  used,  neglecting  the  resistance  of  the  connecting  wires. 
Unless  the  plot  was  very  accurately  made,  the  probable  error 
of  this  result  may  be  a  large  fraction  of  the  battery  resistance. 
This  method  will,  however,  give  accurate  results  when  the  bat- 


5O  RESISTANCE   OF   GALVANOMETER   AND    BATTERY.  O8 

tery  resistance  is  of  about  the  same  magnitude  as  the  average 
resistance  which  was  used  in  the  box  in  taking  measurements 
for  the  plot. 

28.    RESISTANCE    OF    GALVANOMETER    AND 
BATTERY. 

Resistance  of  Galvanometer— Thompson's  Method. 

Put  the  galvanometer  in  the  branch  x  of  the  bridge,  and  let 
the  ordinary  galvanometer  branch  be  replaced  by  a  wire. 
Then  it  is  evident,  first,  that  there  will  always  be  a  current 
through  the  galvanometer  when  the  battery  circuit  is  com- 
pleted; second,  that  if  the  slide  be  so  placed  that  r]x-=plqt 
where  x  is  the  resistance  of  the  galvanometer,  no  current  will 
flow  in  the  wire  cd  on  making  contact  to  the  bridge-wire  by 
means  of  the  slide.  If  the  slide  be  placed  in  any  other  posi- 
tion, there  will  be  a  current  in  cd  on  making  contact,  which 
means  a  redistribution  of  currents  throughout  the  system,  and 
hence  a  change  in  the  current  through  the  galvanometer. 
When  no  current  flows  through  cd  on  making  contact,  there 
is  no  such  redistribution,  and  hence  no  change  in  the  current 
through  the  galvanometer. 

The  method  therefore  consists  in  adjusting  the  slide  until, 
on  making  contact  with  the  slide  on  the  bridge-wire,  there  is 
no  change  in  the  deflection  of  the  galvanometer.  The  current 
from  the  battery  must  be  allowed  to  run  steadily  through  the 
galvanometer,  so  that  the  deflection  may  become  steady  before 
making  a  test;  therefore  a  Daniell  cell  should  be  used.  The 
current  used  will  produce  so  large  a  deflection  that  small 
changes  can  not  be  noticed;  it  can  be  reduced  to  the  proper 
amount  by  connecting  the  two  poles  of  the  cell  directly  by  a 
wire,  to  serve  as  a  shunt 


29]  RESISTANCE    OF    BRIDGE- WIRE.  5! 

Resistance  of  Battery— Mance's  flethod. 

The  galvanometer  and  battery  b  being  in  their  usual  posi- 
tions, let  a  battery  B  be  put  in  the  branch  x.  Then  for  any 
position  of  the  slider,  the  current  through  the  galvanometer 
branch  will  be  the  algebraic  sum  of  the  currents  due  to 
the  batteries  separately.  That  is,  the  effect  of  one  is  super- 
posed on  that  of  the  other.  If  the  slide  be  so  placed  that 
r\x—p\q^  where  x  is  the  resistance  of  the  battery  B,  then  the 
current  through  the  galvanometer  due  to  the  battery  b  is  zero; 
so  that  removing  from  the  system  the  E.  M.  F.  due  to  b  would 
not  change  the  galvanometer  deflection.  It  can  be  shown, 
further,  that  for  this  adjustment,  the  current  through  the 
galvanometer  is  independent  of  both  the  E.  M.  F.  and  the 
resistance  in  the  battery  branch.  (See  Cumming's  Theory  of 
Electricity,  p.  182.)  The  battery  b  may  then  be  replaced  by 
a  key,  and  when  the  slide  is  in  the  proper  position,  making 
and  breaking  the  contact  with  this  key  will  not  change  the 
galvanometer  deflection. 

Thus  the  only  battery  used  is  the  one  whose  resistance  is 
to  be  measured.  Use  the  Daniell  cell  as  before,  and  use  a 
shunt  in  the  appropriate  place  to  reduce  the  galvanometer 
deflection  to  a  proper  amount. 

29.    RESISTANCE   OF   BRIDGE-WIRE. 

The  two  branches 
p  and  q  of  the  bridge 
are  extended  by 
means  of  the  two  un- 
equal known  resist- 
ances A  and  B;  the 

battery  terminals  must  then  be  connected  as  shown;  rand 
x  are  unknown  resistances,  of  about  the  same  magnitude  as 
A  or  B.  Then  for  balance, 


52  RESISTANCE    OF    BRIDGE-WIRE.  1.29 

r/x  =  (A+p)/(B  +  q) 

Now  interchange  A  and  B,  leaving  other  connections  as  before, 
and  find  the  new  positions  for  balance,  for  which 

r/x  =  (B  +  p')/(A  +  q') 

Add  unity  to  each  member  of  the  equations.     This  gives 
r  +  xA  +  p  +  B  +  q  _B  +  p/+A  +  q/ 


_ 
B  +  q  A  +  q' 

Remembering  that  p  +  q  =  p'H-q',  show  that  q  —  q'  =  A  —  B. 

This  is  the  principle  of  Carey  Foster's  method  for  calibrat- 
ing a  bridge-  wire. 

(#.)  For  A,  use  the  i-ohm  spool  of  your  rheostat;  for  B,  a 
short  piece  of  heavy  copper  wire.  A  —  B  will  then  be  the  true 
value  of  the  i-ohm  spool  simply.  For  r  and  x  use  a  meter  of 
No.  25  G.  S.  wire  with  copper  terminals,  the  point  c  being  a 
binding-post  slipped  on  the  wire,  so  that  the  ratio  r>x  may  be 
varied  at  will.  The  wire  on  the  bridge  used  consists  of  two 
equal  meters  of  No.  30  G.  S.  wire,  one  of  which  is  coiled  at 
one  end  of  the  board  and  the  other  stretched  along  the  scale. 
The  measurements  are  of  course  to  be  taken  on  the  stretched 
wire  simply. 

Make  several  determinations  on  different  parts  of  the  wire, 
by  varying  r\x. 

CAUTIONS.  —  Handle  the  wire  with  great  care.  On  account 
of  its  small  diameter  it  is  easily  stretched  or  broken. 

Do  not  press  down  on  the  contact.  The  weight  of  the 
slide  is  sufficient  to  secure  good  contact,  and  any  additional 
pressure  may  injure  the  wire. 

Do  not  slide  the  contact  along  the  wire.  Always  raise  the 
slide  before  moving  it. 

(b.)  Make  another  series  of  determinations,  using  the  2-ohm 
spool  in  place  of  the  i-ohm. 

(c.)  From  all  of  the  results,  calculate  the  average  resistance 
per  cm.  of  the  wire.' 

The  same  wire  is  to  be  used  in  the  next  experiment. 


30]  ELECTROMOTIVE   FORCE.  53 

30.    ELECTROMOTIVE   FORCE. 

Uae  the  same  wire  as  in  the  preceding  experiment.  The 
two  wires,  used  by  the  two  sets  of  observers,  and  the  5o-turn 
coil  of  the  tangent  galvanometer,  should  be  connected  in 
series  with  two  cells  of  the  storage  battery,  giving  an  electro- 
motive force  of  about  four  volts. 

Ask  for  storage  battery  connection. 

Do  not  disturb  the  galvanometer  in  any  way.  If  it  is  out 
of  order  ask  for  assistance.  Its  reduction  factor,  accurately 
determined  for  its  present  position,  is  given. 

Form  a  parallel  circuit  to  the  wire,  consisting  of  one  of  the 
Leclanche  cells  bearing  the  number  of  this  experiment,  and  a 
Wheatstone  galvanometer.  Connect  one  terminal  of  this 
circuit  to  the  sliding  contact,  the  other  to  the  key  at  one  end 
of  the  board,  so  that  depressing  the  key  will  connect  the 
parallel  circuit  to  the  coiled  end  of  the  bridge-wire. 

There  is  an  E.  M.  F.  of  about  2  volts  distributed  along  the 
wire.  Therefore  an  E.  M.  F.  equal  to  that  of  the  Leclanche, 
which  is  less  than  two  volts,  exists  between  the  end  of  the 
wire  and  some  point  along  it.  If  the  sliding  contact  be  placed 
at  this  point,  and  if  the  +  terminal  of  the  Leclanche  be  con- 
nected to  the  same  end  of  the  wire  as  the  +  terminal  of  the 
storage  battery,  then  there  will  be  no  current  through  the 
parallel  circuit.  If  the  Leclanche  be  connected  in  the  opposite 
direction,  however,  there  will  always  be  a  current  through  the 
parallel  circuit,  for  any  position  of  the  slider;  and  this  current 
wjU  obviously  be  always  in  the  same  direction. 

(a.)  In  making  a  test,  set  the  slider  at  the  desired  point  on 
the  wire,  then  depress  the  key  at  the  end  of  the  board  just 
long  enough  to  determine  the  direction  of  the  deflection 
produced.  A  long  contact  will  cause  polarization  of  the 
Leclanche,  and  also  disturb  the  current  in  the  main  circuit 
which  is  being 'used  by  the  other  observers. 


54  ELECTROMOTIVE   FORCE.  [30 

Determine  first  whether  the  Leclanche  is  properly  con- 
nected. Then  adjust  the  slider  until  the  galvanometer  indi- 
cates zero  current,  and  read  its  position.  Read  the  tangent 
galvanometer,  reversing  its  deflection  as  usual.  This  reading, 
as  well  as  the  final  careful  adjustment  of  the  slider,  should  be 
made  when  the  other  pair  of  observers  are  not  making  trial 
adjustments  which  will  disturb  the  main  current. 

($.)  From  Ohm's  law,  calculate  the  E.  M.  F.  between  the 
end  of  the  wire  and  the  sliding  contact,  which  is  equal  to  that 
of  the  Leclanche. 

Make  sure  that  this  method  is  well  understood  before  going 
on,  as  mistakes  with  the  standard  cell  might  be  dangerous. 

(c.)  Repeat  with  the  standard  cell  furnished.  This  is  a  cell 
used  as  a  standard  of  electromotive  force.  Its  constancy 
depends  upon  careful  handling,  and  upon  having  only  minute 
currents  sent  through  it. 

The1  terminal  coming  from  the  upper  end  of  the  cell  is  the 
zinc  or  negative,  corresponding  to  the  zinc  of  the  Leclanche. 
Knowing  also  that  the  E.  M.  F.  is  very  nearly  one  volt,  it 
will  be  possible  by  a  little  calculation  to  set  the  slider  very 
near  to  the  true  point  at  the  start. 

Never  use  a  standard  ceil  without  such  a  preliminary  test 
with  an  ordinary  cell.  If  it  is  impossible  to  approximate 
closely  to  the  true  position  of  the  slider,  a  resistance  of  several 
thousand  ohms  should  be  included  in  series  with  the  cell  to 
protect  it  against  currents  of  dangerous  strength. 

(*/.)  How  could  the  E.  M.  F.  of  any  cell  be  determined  in 
terms  of  the  known  E.  M.  F.  of  the  standard  by  this  method, 
without  using  the  tangent  galvanometer? 

Determine  in  this  way  the  E.  M.  F.  of  a  bichromate  (Grenet) 
cell,  if  one  is  to  be  had.  In  using  this  cell,  the  zinc  should 
be  lowered  into  the  acid  only  while  a  current  is  wanted. 

(e.)  If  there  is  time,  let  each  set  of  observers  determine,  as 
in  (c),  the  E.  M.  F.  of  the  standard  previously  used  by  the 
other. 


31]  BATTERY   RESISTANCE.  55 

A  wire  or  other  resistance  used  as  in  this  experiment  is 
called  a  potentiometer. 

31.  BATTERY   RESISTANCE. 

Let  Ebe  the  E.  M.  F.  of  a  cell  measured  when  it  is  sending 
no  current,  as  in  the  last  experiment.  Let  e  be  the  E.  M.  F. 
measured  between  its  terminals  when  it  is  sending  a  current  c 
through  an  external  resistance  r.  Then  in  this  latter  case,  E 
is  evidently  the  E.  M.  F.  which  sends  the  current  c  through 
the  whole  resistance  in  circuit,  including  that  of  the  battery; 
while  e  is  the  E.  M.  F.  which  sends  the  same  current  through 
the  external  resistance  r. 

Then  calling  the  resistance  of  the  battery  b,  E=c(r+fy, 
e=cr.  From  these  equations  b  can  be  found  if  E  and  e  are 
known,  and  either  c  or  r.  In  this  experiment  we  use  known 
resistances.  Use  a  Daniell  cell  and  resistance  box,  and 
measure  the  E.  M.  F.'s  on  a  potentiometer  supplied  with 
current  from  storage  battery.  The  potentiometer  is  thus 
used  as  a  voltmeter.  Ask  for  storage  battery  connections. 

(a.)  Determine  the  E.  M.  F.  of  the  Daniell  by  comparison 
with  one  of  the  standards  whose  value  was  determined  in 
the  last  experiment.  Use  the  Daniell  first,  remembering 
that  its  E.  M.  F.  is  about  i.i  volts,  in  order  to  get  the  con- 
nections right  and  the  slide  at  about  the  right  point,  for  the 
standard. 

The  result  here  obtained  for  the  Daniell  is  evidently  the 
value  of  E  required. 

.(£.)  Having  the  Daniell  connected  to  the  potentiometer 
as  at  first,  connect  its  terminals  also  to  those  of  the  resist- 
ance box,  using  connecting  wires  of  low  resistance.  The 
potentiometer  may  now  be  used  as  a  voltmeter,  to  determine 
the  E.  M.  F.  existing  between  the  terminals  of  the  cell  while 
it  is  sending  a  current  through  the  known  resistance  in  the 


56  CALIBRATION    OF   A    COPPER-WIRE   THERMOMETER.  1.32 

box.  In  this  way  find  the  values  of  e  for  a  number  of  resist- 
ances, from  10  units  down  to  0.5.  After  each  determination, 
disconnect  the  resistance  box  and  test  the  value  of  E.  The 
reading  with  the  standard  cell  should  be  repeated  at  about 
the  middle  of  the  series,  and  again  at  the  end,  in  order  to 
make  sure  that  the  storage  battery  current  has  remained 
constant  throughout. 

(c.)  Calculate  the  resistance  of  the  battery,  and  also  the 
current  flowing  through  it,  for  each  case. 

(d.)  Plot  the  results  obtained  in  (c\  so  as  to  show  the 
relation  between  the  resistance  of  the  cell  and  the  current 
passing  through  it. 

32.  CALIBRATION    OF   A   COPPER-WIRE 
THERMOMETER. 

The  electrical  resistance  of  all  conductors  changes  with 
change  of  temperature,  and  for  pure  metals  the  change  is 
found  to  be  very  nearly  proportional  to  the  change  of  tem- 
perature. In  this  experiment  the  rate  of  change  of  resistance 
with  temperature  is  to  be  determined  for  a  copper  wire, 
which  may  then  be  used  as  a  thermometer  if  necessary. 
If  the  calibration  curve  be  a  straight  line,  it  is  evident  that 
the  thermometer  can  safely  be  used  for  temperatures  beyond 
the  limits  of  the  mercury  thermometer.  Another  advantage 
of  this  form  of  thermometer  is  that  the  average  temperature 
of  any  body  may  be  determined  very  accurately,  when  it  is 
not  at  a  uniform  temperature  throughout,  by  distributing 
the  copper  wire  uniformly  around  or  through  it. 

The  wire  used  is  No.  32  insulated  copper,  of  about  4  ohms 
resistance,  wound  on  a  sheet  of  mica,  its  terminals  soldered 
to  No.  22  lead-wires. 

A  spool  is  furnished  on  which  are  wound  two  equal 
resistances.  If  these  be  made  the  branches  p  and  q  of  a 


32]     '        CALIBRATION   OF   A    COPPER-WIRE   THERMOMETER.  57 

bridge  arrangement  in  which  the  thermometer  is  the  branch 
x  and  the  bridge-wire,  measured  from  one  end  to  the  sliding 
contact,  is  the  branch  r,  then  it  is  evident  that  if  r  can  be 
adjusted,  by  means  of  the  sliding  contact,  so  as  to  give  no 
current  in  the  galvanometer  circuit,  the  resistance  in  r  is 
exactly  equal  to  that  of  the  thermometer  wire,  and  its  lead- 
wires.  In  order  to  eliminate  the  effect  of  these  leads,  a  pair 
of  "  compensating  leads,"  consisting  of  a  simple  loop  of  wire 
of  the  same  size  and  length  as  the  thermometer  leads,  is 
wound  with  them,  so  that  the  resistance  of  the  two  pairs  of 
leads  will  be  the  same  under  all  circumstances.  If  these 
compensating  leads  be  put  in  r,  in  series  with  the  bridge- 
wire,  the  resistance  of  the  bridge-wire  in  r  will  then  be 
equal  to  that  of  the  No.  32  wire  of  the  thermometer  simply. 

The  two  25  cm.  lengths 
on  the  bridge  need  not 
be  used.  The  50  cm. 
along  the  scale  is  to  be 
extended  by  a  spool  k 
equal  in  resistance  to  250 
cm.  of  the  wire,  so  that 
the  actual  length  of  wire 
up  to  the  sliding  contact  is  the  length  on  the  scale  added  to 
250. 

In  setting  up  the  arrangement,  notice  that  the  point  a  is 
the  junction  of  one  thermometer  lead  with  one  end  of  the 
spool  carrying/  and  q,  b  is  the  binding-screw  on  the  sliding 
contact,  to  which  the  other  end  of  the  spool  is  attached,  c  is 
the  junction  of  the  second  thermometer  lead  with  one  of  the 
compensating  leads,  the  other  being  connected  to  the  coil  k 
at  the  end  of  the  bridge,  and  d  is  the  junction  of  p  and  q. 

The  thermometer  leads  are  tied  with  a  piece  of  twine  near 
the  ends,  to  distinguish  them  from  the  compensating  leads. 

(a.)  Use  a  kettle  for  a  water-bath  in  which  to  immerse  the 


5#  STUDY   OF   POLARIZATION   EFFECTS.  [33 

copper  thermometer  and  the  attached  mercury  thermometer. 
Record  the  position  of  the  slider  for  temperatures  at  intervals 
of  about  5°  through  the  range  from  20°  to  40°.  For  each 
reading  keep  the  temperature  steady,  within  o°.i,  and  the 
bath  well  stirred  until  the  position  of  the  slider  for  balance 
remains  constant,  showing  that  the  wire  has  attained  the 
same  temperature  as  that  indicated  by  the  thermometer. 
Take  care  to  make  the  contacts  in  the  battery  circuit  as 
short  as  possible,  in  order  to  avoid  heating  the  thermometer 
wire  by  the  testing  current. 

Take  a  similar  series  of  descending  readings,  at  about  the 
middle  points  of  the  temperature  intervals  previously  used. 

(&)  Plot  the  observations,  using  temperatures  as  abscissae 
and  changes  in  position  of  slider  as  ordinates,  each  on  as 
large  a  scale  as  the  paper  will  permit.  The  calibration  curve 
for  the  mercury  thermometer  used  will  be  furnished  on 
application.  Since  the  length  of  the  bridge-wire  is  always 
proportional  to  the  resistance  of  the  thermometer  wire, 
changes  in  position  of  the  slider  will  be  proportional  to 
corresponding  changes  in  that  resistance;  and  a  knowledge 
of  the  actual  resistance  of  the  wire  is  unnecessary. 

(c.)  Calculate  the  coefficient  of  increase  of  resistance  per 
degree,  in  terms  of  the  resistance  at  o°. 

(</.)  Calculate  the  temperature  at  which  the  resistance  of 
the  wire  would  become  zero.  What  must  you  assume? 

33.  STUDY   OF   POLARIZATION   EFFECTS. 

Use  the  same  voltmeter  as  in  Exp.  23,  and  see  that  the 
proper  resistance  coil  is  connected  in  series  with  it.  Also 
use  a  Leclanche  cell  which  bears  the  number  of  this  experi- 
ment, and  no  other. 

In  this  experiment  the  closing  of  a  key  produces  a  deflec- 
tion of  the  voltmeter  corresponding  to  an  E.  M.  F.  which  is 


33]  STUDY   OF   POLARIZATION   EFFECTS.  59 

changing  rather  rapidly,  so  that  a  reading  must  be  obtained 
as  soon  as  possible.  The  needle,  however,  does  not  come  to 
rest  until  after  several  vibrations;  so  that  the  best  plan  is 
to  read  three  consecutive  turning-points  (the  first  three  if 
possible)  and  calculate  from  them  the  resting-point,  as  was 
done  with  the  balance.  A  little  practice  at  this  before  begin- 
ning the  experiment  will  be  of  advantage. 

(#.)  Connect  the  terminals  of  the  cell  directly  with  the 
terminals  of  the  voltmeter,  and  so  get  a  reading  which  is 
proportional  to  the  total  E.  M.  F.  E  of  the  cell.  It  may  be 
necessary  to  move  the  voltmeter  coils  farther  away  from  the 
needle  in  order  to  get  a  deflection  which  can  be  read  on  the 
scale.  This  deflection  should  be  at  least  3  cm.  from  the 
end  of  the  scale,  for  convenience  in  reading  turning-points 
later  on. 

Throughout  the  succeeding  series  of  observations  the  volt- 
meter should  remain  connected  to  the  terminals  of  the  cell, 
except  that  it  should  be  disconnected  at  convenient  intervals 
to  check  the  zero  reading  on  the  scale. 

(^.)  Connect  the  terminals  of  the  cell  through  a  spool  of  2 
ohms  resistance  and  a  break-circuit  key  (one  which  breaks 
connection  instead  of  making  it  on  depressing  the  lever). 
Complete  this  parallel  circuit  at  a  noted  time  and  immediately 
read  the  voltmeter  deflection  e  by  means  of  three  turning- 
points.  Be  careful  not  to  connect  the  cell  in  such  a  way  as 
to  send  a  current  through  it  until  ready  to  begin  observations. 

(c^)  At  the  end  of  five  minutes  read  e  again,  disconnect  the 
resistance,  by  means  of  the  break-circuit  key,  and  as  quickly 
as* possible  read  E,  by  means  of  three  turning-points.  This 
reading  must  be  taken  quickly  because  E,  which  has  been 
reduced  by  polarization  caused  by  the  passage  of  the  current, 
immediately  begins  to  return  to  its  former  value. 

(d.)  Re-connect  the  resistance,  and  note  the  time  at  which 
e  again  has  the  same  value  as  that  read  in  (c).  Count  from 


60  WRITTEN   EXERCISE— ELECTRICITY.  [34 

this  another  5 -minute  interval,  then  repeat  the  readings  as 
in  (c).  By  this  means  the  readings  obtained  are  very  nearly 
the  same  that  they  would  have  been  if  no  time  had  been  lost 
in  taking  them,  so  that  the  current  could  flow  continuously. 

Continue  the  readings  in  this  way  until  four  periods  of  $ 
minutes  each  have  been  covered. 

(>.)  Leaving  the  resistance  disconnected,  take  readings  of 
E  at  5-minute  intervals  for  about  45  minutes. 

At  some  time  during  this  interval,  secure  if  possible  the 
Leclanche  cell  whose  E.  M.  F.  was  determined  in  Exp.  30,  and 
by  getting  the  voltmeter  deflection  corresponding  to  its  E. 
M.  F.,  obtain  a  factor  for  reducing  your  readings  to  volts. 

Why  could  not  a  standard  cell,  whose  resistance  is  about 
1,000  ohms,  be  used  for  this  purpose? 

(/".)  Plot  a  curve  with  times  as  abscissae  and  corresponding 
values  of  E  as  ordinates,  for  the  whole  time  of  the  experiment. 
Remember  that  during  the  first  part  the  5-minute  intervals  only 
are  to  be  counted. 

Calculate,  as  in  the  last  experiment,  the  resistance  of  the 
ceil  from  each  pair  of  values  of  e  and  E,  tabulate  the  results, 
and  show  that  for  this  purpose  the  scale-readings  correspond- 
ing to  E  and  e  may  be  used  without  knowing  their  values 
in  volts. 

34.    WRITTEN   EXERCISE— ELECTRICITY. 

(i.)  Arrange  the  details  of  a  method  for  calibrating  the 
sensitive  galvanometer  whose  resistance  was  determined  in  Exp. 
28,  using  your  tangent  galvanometer  to  determine  the  strengths 
of  the  currents  used,  and  dividing  these  currents  so  as  to  send 
known  fractions  through  the  sensitive  galvanometer. 

The  curve  obtained  should  read  amperes  directly.  Will  this 
probably  be  a  tangent  curve? 

(2.)  Having  given  this  calibration  curve  for  the  sensitive 
galvanometer,  show  how  you  could  use  it  as  a  voltmeter. 


35-36]  INDEX    OF    REFRACTION.  6 1 

(3.)  Show  that  the  scale  readings  in  Exp.  23  are  only  approx- 
imately proportional  to  the  tangents  of  the  corresponding 
angles  of  deflection,  using  the  simple  law  of  reflection  at  a 
plane  mirror. 

(4.)  Show  that  the  Carey  Foster  method  could  be  used 
to  find  the  difference  between  two  nearly  equal  resistances, 
A  and  B. 

(5.)  Show  how  to  modify  the  copper- wire  thermometer 
apparatus  in  32  so  that  it  will  be  capable  of  reading  to  o°.ooi 
through  a  range  of  5°. 

35-36.    INDEX  OF  REFRACTION.     (Two   Periods.) 

A  small  piece  of  plate  glass  is  set  vertically,  with  its  upper 
edge  horizontal,  at  the  center  of  a  rotating  table.  The  arm 
to  which  the  glass  is  directly  attached,  by  means  of  soft  wax, 
can  be  turned  independently  of  the  table,  but  moves  with 
the  table  when  the  latter  is  rotated.  A  millimeter  scale  is 
placed  with  its  divisions  vertical,  in  such  a  position  that  when 
a  telescope  at  the  same  height  as  the  upper  edge  of  the  glass 
is  focused  on  the  scale,  its  divisions  will  be  seen  partly 
through  £he  glass  plate,  and  partly  above  its  upper  edge. 
The  scale  should  be  normal  to  the  line  of  sight,  with  its 
middle  division,  marked  zero,  behind  the  middle  of  the  glass 
plate.  Set  the  eye- piece  of  the  telescope  so  that  the  cross- 
wires  are  at  an  angle  of  45°  with  the  vertical,  and  with  their 
intersection  on  the  zero  line  above  the  glass.  The  image  seen 
in  the  telescope  will  then  consist  of  two  parts,  one  formed 
by  light  coming  through  the  glass  plate,  the  other  by  light 
coming  through  the  air  above  the  plate.  These  two  images 
will  appear  to  coincide  when  the  plate  is  normal  to  the  line 
of  sight ;  but  when  the  plate  is  rotated  in  either  direction 
about  a  vertical  axis,  one  portion  of  the  image  is  displaced 
with  reference  to  the  other.  For  a  certain  angle  of  rotation, 


62 


INDEX    OF    REFRACTION. 


[35-36 


the  lines  of  the  scale  will  again  appear  to  be  continuous, 
instead  of  broken.  This  evidently  means  that  one  portion  of 
the  image  has  been  displaced  with  reference  to  the  other 
by  I  mm.  Continuing  the  rotation,  the  lines  of  the  scale 
will  again  appear  to  be  continuous  for  displacements  of 
2,  3.  etc.,  mm. 

In  the  figure,  PBOT  repre- 
sents the  path  of  the  pencil, 
from  the  scale  to  the  tel- 
escope, by  which  part  of 
one  scale  division  is  seen 
through  the  glass;  QAOT, 
that  by  which  part  of  the 
adjacent  division  is  seen  above 
the  glass.  The  two  paths  co- 
incide after  leaving  the  glass, 
so  that  the  two  parts  of  the 
two  lines  appear  to  coincide 
and  form  one  line  in  the  image. 
/  is  thickness  of  plate. 

d  is  distance  which  one  image  is  displaced  with  reference  to 
the  other. 

i  is  angle  of  incidence  (or  emergence). 
r  is  angle  of  refraction. 

The  index  of  refraction  is  to  be  obtained  from  the  formula 
n  =  s'm  z/sin  r. 

All  of  the  above  quantities  will  be  determined  directly, 
except  r,  which  can  be  obtained  from  the  relation  tan  r=tan  i- 
(d/t  cos  i).  Show  the  truth  of  this  relation  from  the  figure. 
(a.)  To  determine  the  values  of  i  corresponding  to  displace- 
ments of  i,  2,  3,  and  4  mm.  Either  the  arm  to  which  the 
glass  is  attached,  or  the  whole  table,  can  be  rotated  by  the 
observer  at  the  telescope  by  means  of  a  wooden  rod  with  a 
peg  at  the  end,  which  can  be  set  in  one  of  the  holes  made  for 
the  purpose. 


35-36]  INDEX   OF    REFRACTION.  63 

Although  i  may  be  observed  directly,  a  better  result  will  be 
obtained  by  measuring  21.  To  do  this,  set  the  plate  so  that  a 
displacement  of  I  mm.  in  one  direction  is  obtained,  then  read 
the  angle  through  which  the  table  must  be  rotated  to  cause 
an  equal  displacement  in  the  other  direction. 

It  will  soon  be  seen  that  the  accuracy  of  a  single  setting  is 
not  very  great,  so  that  it  is  desirable  to  average  a  large  number 
of  settings.  This  can  be  done  without  reading  each  setting, 
as  follows:  Set  the  circle  so  that  the  index  reads  zero ;  set  the 
glass,  by  means  of  the  movable  arm,  to  give  a  displacement 
of  I  mm.;  turn  the  table  until  an  equal  displacement  in  the 
opposite  direction  is  secured;  leaving  the  table  in  this  position, 
turn  the  glass  back  to  its  original  position  by  means  of  the 
movable  arm;  again  turn  the  glass  into  its  second  position  by 
means  of  the  table.  It  is  evident  that  each  time  the  operation 
is  repeated,  the  table  is  turned  through  an  angle  2t;  and  thus 
any  number  of  observations  may  be  automatically  added. 

Make  a  series  of  determinations  in  this  way  until  the  sum 
of  the  angles  reaches  three  or  four  times  360°.  The  final 
reading  should  be  taken  at  a  point  as  near  to  the  zero  of  the 
circle  as  possible,  which  will  reduce  the  effect  of  errors  in  the 
graduation  or  mounting  of  the  circle.  The  second  observer 
should  note  the  number  of  settings  made,  and  the  number  of 
times  the  zero  of  the  circle  passes  the  index.  Calculate  the 
mean  value  of  i  for  the  series. 

In  this  way  let  each  observer  make  a  determination  of  i  for 
each  displacement  of  i,  2,  3,  and  4  millimeters. 

(£.)  To  determine  /,  by  means  of  the  OPTICAL  LEVER. 

The  optical  lever  is  a  short  lever  of  brass  resting  by  three 
points  on  a  bed-plate  of  brass.  Two  of  the  points,  at  one  end, 
rest  in  minute  conical  depressions  in  the  bed-plate,  which  keep 
them  in  definite  position,  and  serve  as  the  fulcrum  of  the  lever. 
The  perpendicular  distance  from  the  third  point  to  the  line 
joining  these  two  points  is  the  effective  length  of  the  lever. 


64  INDEX    OF    REFRACTION.  [35~36 

The  distance  through  which  the  third  point  is  raised  when 
any  object  is  placed  under  it,  or  the  thickness  of  the  object,  is 
to  be  calculated  from  the  length  of  the  lever  and  the  angle 
through  which  it  is  rotated. 

This  angle  is  to  be  measured  by  means  of  a  modification  of 
the  telescope  and  scale  method.  A  scale  is  set  in  a  vertical 
position  at  a  distance  of  75  to  100  cm.  in  front  of  the  mirror 
which  is  attached  to  the  lever.  A  slider  attached  to  the  scale 
is  pierced  with  a  small  round  hole,  and  has  on  the  side  toward 
the  mirror  a  horizontal  line  whose  direction  passes  through 
the  center  of  the  hole.  Adjust  the  relative  positions  of  scale 
and  lever  so  that  when  the  slider  is  at  the  proper  height,  the 
reflected  image  of  the  horizontal  line  can  be  seen  in  the  mirror, 
on  looking  through  the  hole.  Adjust  the  slider  until  the 
image  of  the  line  appears  to  bisect  the  small  round  ink-spot 
near  the  center  of  the  mirror.  The  adjustment  can  be  made 
more  closely  if  the  eye  be  held  a  short  distance  from  the  hole, 
and  so  that  the  spot  on  the  mirror  appears  to  be  exactly  in 
the  center  of  the  hole.  It  is  now  evident  that  the  normal  to 
the  mirror,  at  the  center  of  the  spot,  passes  through  the  center 
of  the  hole. 

The  scale  should  now  be  vertical,  and  the  normal  to  the 
mirror  horizontal,  that  they  may  form  two  sides  of  a  right 
triangle.  Adjust  the  former  by  sighting  on  window  casings 
or  other  convenient  vertical  lines.  Test  the  latter  by  measur- 
ing from  the  table  the  height  of  the  line  on  the  slider  and  of 
the  spot  on  the  mirror,  and  make  any  necessary  adjustment 
by  means  of  the  screw  to  which  the  third  point  is  attached. 
Having  these  adjustments  made,  read  the  position  of  the 
slider  by  means  of  the  index. 

The  thickness  of  the  plate  is  only  needed  over  the  portion 
actually  used  in  (a).  The  settings  made  should  therefore  be 
confined  to  this  portion,  and  should  cover  it  with  some 
attempt  at  uniformity,  so  that  the  mean  thickness  of  the  plate 
as  used  will  be  determined. 


37-38]  SPHEROMETER.  65 

Carefully  lift  the  third  point  of  the  lever,  and  insert  the 
plate,  letting  it  rest  on  the  three  projections  from  the  bed- 
plate. See  that  the  two  points  at  the  other  end  of  the  lever 
have  not  been  disturbed  from  their  position,  then  adjust  and 
read  the  slider  as  before.  Make  several  settings  of  the  slider 
for  different  points  over  the  desired  area  of  the  plate,  then 
remove  the  latter  and  check  the  zero  reading. 

Measure  the  horizontal  distance  from  the  line  on  the  slider 
to  the  face  of  the  mirror,  and  calculate  the  mean  angle  through 
which  the  lever  has  been  rotated. 

Repeat  for  at  least  two  other  distances  of  the  scale  from 
the  mirror.  Note  that  if  the  distance  be  made  just  I  meter, 
the  difference  between  the  two  readings  of  the  slider  gives 
the  tangent  of  the  angle  directly. 

Calculate  the  mean  angle  from  the  whole  series. 

To  find  the  length  of  the  lever,  lay  it  on  the  table  with  the 
points  up,  and  lay  on  them  a  half-millimeter  scale  ruled  on 
glass,  ruled  side  down.  Set  the  two  points  at  one  end  of  the 
lever  in  the  long  zero  division,  and  read  the  position  of  the 
third  point  as  closely  as  possible  with  a  magnifying  glass. 
This  evidently  gives  the  desired  distance,  since  the  zero  divi- 
sion is  perpendicular  to  the  length  of  the  scale. 

This  gives  all  the  data  needed  for  calculating  the  thickness 
of  the  working  portion  of  the  glass  plate. 

(c.)  Calculate  the  index  of  refraction  of  the  glass,  from  each 
pair  of  values  of  i  and  d  obtained  in  (a\  and  find  the  mean  of 
these  results. 

37-38.   SPHEROMETER.     (Two    Periods.) 

* 

Radii  of  Curvature  of  Lens  Surfaces. 

PRELIMINARY. — Testing  vernier  calipers. 
Secure  the  billiard  ball  whose  volume  was  determined  in 
Exp.  5,  and  from  that  volume  find  the  mean  diameter  of  the 


66  SPHEROMETER.  [37~38 

ball  (Whiting's  Tables,  3H).  This  gives  a  standard  of  length 
whose  value  is  independent  of  any  direct  measurement  on  a 
scale.  (See  Whiting,  p.  72.)  By  measuring  the  diameter  of 
the  ball  with  the  vernier  calipers,  the  scale  of  the  calipers  may 
be  compared  with  this  standard. 

As  the  ball  is  not  a  perfect  sphere,  it  will  be  necessary  to 
measure  a  number  of  diameters  in  different  directions,  uni- 
formly distributed,  in  order  to  get  a  result  which  may  be 
compared  with  the  diameter  calculated  from  the  volume.  In 
order  to  get  a  uniform  distribution  of  the  measurements,  draw 
lightly  in  pencil  on  the  ball  two  great  circles  at  right  angles, 
and  a  third*  perpendicular  to  both.  The  intersections  of  these 
will  give  three  diameters,  and  points  on  each  circle  midway 
between  the  intersections  will  give  six  more.  Four  others 
may  be  taken  connecting  the  centers  of  the  triangles  into 
which  the  surface  has  been  divided. 

Only  one  pair  of  calipers  is  available,  so  that  while  one  set 
of  observers  is  using  it,  the  other  should  be  taking  the  spher- 
ometer  readings  in  (a)  and  (b). 

In  using  the  calipers,  great  care  should  be  taken  not  to 
strain  the  jaws.  Ask  for  special  directions  as  to  the  necessary 
precautions.  Make  a  coarse  adjustment  by  taking  hold  of 
the  movable  jaw  (not  the  small  slide  attached  to  it)  and  sliding 
it  along  the  scale.  Then  clamp  the  small  slide,  and  make  the 
final  adjustment  by  means  of  the  slow-motion  screw,  turning 
it  until  the  object  is  touched  lightly  by  both  jaws,  but  not  held 
firmly  between  them. 

To  set  on  a  given  diameter  of  the  ball,  hold  it  with  one 
marked  end  of  the  diameter  against  the  fixed  jaw,  just  inside 
the  end.  Then  adjust  the  movable  jaw  until,  on  moving  the 
ball  slightly  back  and  forth,  it  just  grazes  the  corresponding 
point  on  this  jaw.  Never  clamp  the  ball  so  tightly  that  its 
weight  can  be  supported  by  the  calipers.  The  accuracy  of  the 
results  will  depend  largely  on  the  uniformity  of  the  pressure 
with  which  these  settings  are  made. 


37-38]  SPHEROMETER.  6/ 

After  making  the  settings  on  the  ball,  close  the  movable 
jaw  up  against  the  fixed  jaw  with  as  nearly  as  possible  the 
same  pressure  used  on  the  ball,  and  determine  the  zero  error. 
If  any  is  found,  allow  for  it  on  the  measurements  taken. 

Find  the  factor  by  which  the  mean  diameter  in  terms  of  the 
caliper  scale  must  be  reduced  to  the  calculated  value,  and  if 
this  factor  differs  from  unity  by  an  appreciable  amount,  use  it 
in  future  work. 

(a.)  Pitch  of  spherometer  screw. 

A  rotating  arm  is  provided  which  slips  over  the  point  of 
the  spherometer  screw.  By  means  of  this,  adjust  the  upper 
brass  ring  so  that  its  axis  coincides  with  the  axis  of  rotation 
of  the  screw,  and  clamp  it  in  that  position  by  means  of  the 
small  clamp-screws  underneath. 

Fasten  a  small  piece  of  plate  glass  to  a  larger  piece  with  soft 
wax,  making  sure  that  the  surfaces  are  clean  and  accurately 
in  contact.  Set  the  plate  on  the  ring  of  the  spherometer,  with 
the  attached  piece  downward,  and  while  holding  it  down  with 
one  hand,  turn  the  screw  upward  by  means  of  the  milled  head 
underneath  the  disk.  Never  turn  the  screw  by  means  of  the 
disk  itself.  The  screw  turns  very  easily,  so  that  the  moment 
of  contact  can  be  felt,  and  care  should  be  taken  not  to  strain 
the  screw  by  turning  beyond  this  point.  Find  the  number  of 
turns  of  the  screw  corresponding  to  the  thickness  of  the  piece 
of  glass  at  two  selected  points  near  its  edges,  by  setting  the 
screw  first  on  the  selected  points,  and  then  on  the  plate  when 
the  piece  has  been  removed. 

Measure  the  thickness  of  the  piece  at  each  of  the  two  points, 
making  several  settings  of  the  calipers  in  each  case.  Calcu- 
late the  pitch  of  the  screw,  or  the  distance  in  centimeters 
which  its  point  advances  for  one  revolution. 

Also  measure  with  the  calipers  the  internal  diameter  of  the 
upper  ring  on  the  spherometer,  and  check  the  measurement 
with  the  glass  scale  used  in  the  last  experiment.  The  radius 


68  FOCAL  LENGTH  OF  LENS.  [39 

of  this  ring,  which  is  the  perpendicular  distance  from  the  axis 
of  the  screw  to  the  inner  edge  of  the  ring,  is  called  the 
"  Span"  s  of  the  spherometer. 

When  you  have  finished  with  the  calipers  wipe  them  care- 
fully with  a  dry  cloth  before  putting  them  in  the  case,  to  guard 
against  rust. 

(b.y  Set  the  ball  in  the  ring  of  the  spherometer,  and  bring 
up  the  screw  as  in  (a)  against  its  lower  surface.  In  this  way 
make  a  setting  on  each  of  the  previously  marked  points,  in 
order  to  find  the  mean  distance  d  from  the  point  of  the  screw, 
when  thus  resting  against  the  surface  of  the  ball,  to  the  plane 
of  the  upper  surface  of  the  ring. 

To  calculate  the  diameter  of  the  ball  from  these  data,  con- 
sider a  section  of  the  ball  made  by  a  plane  through  the 
common  axis  of  the  screw  and  the  ring.  In  this  figure,  if  D 
be  the  vertical  diameter  of  the  ball,  show  that 

s2  =  d(D-d) 

If  the  value  of  D,  calculated  from  this  formula,  agrees  closely 
with  the  value  first  obtained  from  the  volume,  it  is  evidence 
that  the  constants  of  the  spherometer  have  been  accurately 
determined  in  terms  of  your  absolute  length  standard,  and 
that  the  method  of  calculation  is  well  understood. 

(<;.)  Find  the  radius  of  curvature  of  each  surface  of  the  given 
lens  from  a  number  of  readings  on  different  parts  of  the 
surface. 

The  same  lens  is  to  be  used  in  several  succeeding  experi- 
ments. When  not  in  use  it  should  be  kept  in  the  rack  pro- 
vided for  the  purpose.  Be  careful  at  all  times  to  avoid 
scratching  the  surfaces  of  the  lens  in  using. 

39.     FOCAL  LENGTH  OF  LENS. 

The  focal  length  of  the  lens  is  first  to  be  determined  by  the 
method  of  conjugate  foci.  Fix  the  lens  on  one  of  the  slides 


391  FOCAL  LENGTH  OF  LENS.  69 

of  the  optical  bench,  which  is  fitted  to  hold  it.  A  large  screen., 
at  one  end  of  the  bench  is  fitted  with  fine  threads  stretched 
across  a  small  hole,  to  serve  as  objects.  The  similar  threads 
stretched  across  the  hole  in  the  small  screen  on  the  second 
slide  of  the  bench  are  used  to  locate  the  image,  formed  by  the 
lens,  of  the  object. 

Move  the  two  slides  as  near  to  the  large  screen  as  the 
bench  will  allow,  and  adjust  the  two  screens  so  that  the  cen- 
ters of  the  openings  and  the  center  of  the  lens  shall  be  on  the 
same> straight  line  parallel  to  tl.^  bench.  The  centers  will  then 
remain  on  this  line  as  the  slides  are  moved.  This  adjustment 
can  be  made  well  enough  by  eye,  or  by  rough  measurement 
from  the  bench. 

(a.)  For  each  observation,  the  distance  of  the  lens  from 
each  screen  is  needed;  and  it  is  evident  that  these  distances 
can  not  be  directly  obtained  by  reading  the  positions  of  the 
slides  on  the  meter  scale.  To  provide  the  means  of  getting 
the  true  distances  from  any  set  of  readings,  set  the  slides  at 
any  convenient  positions,  read  their  indices,  and  measure  with 
a  beam  caliper  the  distance  from  each  set  of  threads  to  the 
corresponding  lens  surface.  Take  three  sets  of  measurements 
for  different  positions  of  the  slide. 

The  lens  is  assumed  to  be  a  thin  lens,  or  one  whose  thick- 
ness may  be  neglected.  The  approximation  arising  from  this 
assumption  will  be  less,  for  the  double  convex  lens  used,  if 
the  measurements  are  made  from  the  center  instead  of  from 
the  surfaces.  The  thickness  of  the  lens,  for  this  purpose,  can 
be  measured  with  sufficient  accuracy  as  follows:  Put  two 
small  wooden  blocks  on  the  two  slides,  setting  them  so  that 
the  adjacent  surfaces  are  in  contact,  and  read  the  distance 
between  the  indices.  Then  separate  the  slides,  insert  the  lens 
between  the  blocks,  and  read  the  distance  again. 

(£.)  Bring  the  image  screen  back  nearly  to  the  opposite  end 
of  the  bench  from  the  object,  and  bring  the  lens  toward  it  until, 


7O  FOCAL  LENGTH  OF  LENS.  [39 

on  looking  toward  the  lens  from  about  30  cm.  behind  the 
image  screen,  the  image  of  the  object  is  seen  through  the 
hole  in  the  screen.  Adjust  the  lens  then  until  the  image 
seems  to  be  in  the  same  plane  as  the  threads  in  the  screen^ 
tested  by  seeing  that  there  is  no  parallax  between  the  two  on 
moving  the  eye  from  side  to  side. 

To  make  a  closer  adjustment,  hold  a  magnifying  glass  so 
that  the  threads  are  clearly  seen.  If  the  image  be  not  in  the 
same  plane,  it  will  not  be  clearly  in  focus  at  the  same  time, 
and  any  remaining  parallax  will  be  magnified.  Find  the  object, 
distance  u. 

Leaving  the  lens  where  it  is,  so  that  u  will  remain  the  same, 
move  the  image  screen  and  then  readjust  it.  Practice  in  this 
way  until  settings  can  be  made  which  agree  within  I  or  2  mm.r 
then  from  the  mean  of  several  settings  find  the  image  distance  v^ 

Move  the  lens  toward  the  object  until,  by  rough  trial,  v  is 
found  nearly  equal  to  u.  Then  determine  n  and  v  as  before. 

(c.)  Lay  off  on  co-ordinate  paper  the  first  pair  of  values  of 
u  and  v  along  the  axes  of  X  and  Y  respectively,  on  a  scale  as 
large  as  possible,  and  connect  the  two  points  by  a  straight  line- 
Do  the  same  for  the  other  pair  of  values.  Draw  the  lines 
accurately,  with  a  fine  pointed  hard  pencil. 

Write  the  equation  of  either  of  the  lines,  and  show  that  if 

i/v+  i/u=  i/f, 

the  point,  each  of  whose  co-ordinates  is/,  will  lie  on  this  line, 
and  hence  will  be  the  point  of  intersection  of  two  such  lines. 
It  will  of  course  also  lie  on  the  line  making  an  angle  of  45° 
with  each  axis.  Draw  this  45°  line.  Then  the  best  value  of 
/  will  be  found  by  locating  the  point  which  represents  the 
average  of  the  points  in  which  the  ?/,  v  lines  intersect  the  45° 
line. 

Find  several  additional  pairs  of  values  of  u  and  v,  choosing 
them  so  that  the  resulting  lines  will  cover  as  wide  a  range  in 
direction  as  possible,  and  use  the  average  from  all.  For  some 


4°]  FOCAL  LENGTH  OF  LENS.  /I 

of  them,  v  should  be  made  greater  than  u,  so  as  to  give  a 
magnified  image. 

If  this  magnified  image  is  indistinct,  so  that  it  is  difficult  to 
make  an  accurate  setting,  cut  a  hole  about  I  cm.  in  diameter 
in  a  piece  of  paper,  and  put  it  in  front  of  the  lens  so  that  only 
the  central  portion  of  the  lens  will  be  effective  in  forming  the 
image. 

(d.~)  Move  the  opening  in  the  piece  of  paper  out  near  one 
edge  of  the  lens,  so  that  the  image  is  now  formed  by  a  portion 
of  the  lens  away  from  the  center.  Determine  the  focal  length 
as  in  (c)  for  this  part  of  the  lens.  Then  explain  the  advan- 
tage of  a  diaphragm  with  a  small  central  opening  for  securing 
a  distinct  and  definite  image ;  and  see  also  that  in  looking  at  the 
image  with  the  eye,  the  image  formed  by  any  desired  portion 
of  the  lens  can  be  seen  by  holding  the  eye  in  the  appropriate 
position.  The  office  of  the  diaphragm  in  this  case  is  merely 
to  make  it  certain  that  the  eye  sees  the  desired  image  and  no 
other. 

40.  FOCAL  LENGTH  AND  INDEX  OF  REFRACTION 
FOR  RED  AND  BLUE  LIGHT. 

Since  the  focal  length  of  a  lens  depends  upon  its  index  of 
refraction,  which  varies  for  different  wave-lengths,  it  is  evident 
that  the  value  of/  determined  in  the  last  experiment  was  an 
average  value  for  the  brighter  portion  of  the  spectrum  of  the 
light  used.  In  order  to  determine  f  for  two  fairly  definite 
colors  of  the  spectrum,  red  and  blue,  put  in  front  of  the  object 
threads  a  piece  of  blue  glass,  which,  it  will  be  remembered, 
transmits  some  red  light  as  well  as  blue.  Illuminate  the 
threads  by  a  kerosene  lamp  placed  behind  them.  This  light, 
being  richer  in  red  as  compared  with  blue  than  ordinary  day- 
light, will  give  more  nearly  equal  brightness  in  the  red  and 
blue  transmitted  by  the  glass.  We  will  now  have  two  images 


/2  INDEX    OF    REFRACTION    OF   A    LIQUID.  [41 

formed  by  the  lens — one  by  red  and  one  by  blue  light.  On 
setting  the  image-screen  and  applying  the  magnifying  glass  it 
will  be  seen  that  there  are  two  positions  of  the  screen  which 
give  a  clear  image.  At  one  position  the  threads,  and  the 
margin  of  the  aperture  in  the  screen,  appear  red  and  the 
spaces  bluish;  at  the  other  the  threads  and  margin  appear  blue 
and  the  spaces  reddish.  Decide  as  to  which  image  is  formed 
by  blue  and  which  by  red  light,  and  explain  the  color 
phenomena  in  the  two  cases. 

For  convenience,  the  magnifying  glass  is  attached  to  the 
image-screen,  so  that  it  can  be  focused  on  the  thread  and 
moved  with  the  screen,  or  turned  one  side  for  a  rough 
setting. 

(«.)  Determine,  by  the  method  of  the  last  experiment,  the 
focal  length  of  the  lens  for  red  and  for  blue  light.  Locate 
both  images  for  each  position  of  the  lens,  and  so  carry  on 
both  series  at  once. 

($.)  From  the  radii  of  curvature  of  the  lens  surfaces,  and/" 
as  found  in  the  last  experiment,  calculate  its  index  of  refrac- 
tion for  the  light  there  used.  In  the  same  way,  from  the  data 
obtained  in  this  experiment,  calculate  its  index  of  refraction 
for  red  and  for  blue  light. 

41.    INDEX    OF    REFRACTION   OF   A    LIQUID. 

(a.)  Inclose  a  little  water  between  a  piece  of  plate  glass  and 
the  lens  previously  used,  keeping  it  in  place  by  a  cell  made 
from  a  piece  of  sheet  rubber  with  a  round  hole  cut  from  its 
center.  Mount  the  whole  on  the  optical  bench,  and  determine 
the  focal  length  of  the  combination,  for  daylight.  Use  the 
following  method:  — 

For  a  given  position  of  the  lens,  let  u  +  v=/t  the  distance 
between  the  screens.  Then  leaving  the  screens  fixed,  move 
the  lens  to  another  position,  such  that  uv  —  v  and  ^  =  u.  If  the 


42]  INDEX    OF    REFRACTION    OF    LENS.  73 

first  image  was  reduced,  the  second  will  be  enlarged;  and  the 
distance  through  which  the  lens  was  moved  (read  directly  on 
the  scale)  is  v  —  vl  =  v  —  u=a.  Show  then  that  /=  (/2  —  a*)  /  4!. 
This  does  not  involve  any  measurements  to  the  lens  surfaces, 
which  might  be  awkward  in  this  case. 

The  plate  glass  used  is  of  the  same  piece  as  that  whose 
index  of  refraction  was  determined  in  Exp.  36.  If  u  above  in- 
clude this  plate  glass,  show  how  to  correct  it  so  as  to  get  the 
value  which  u  would  have  in  air;  and  show  that  this  correction 
should  be  applied  to  /  as  measured  above. 

(&)  Calculate  the  focal  length  and  index  of  refraction  of  the 
liquid  lens. 

42.    INDEX    OF    REFRACTION    OF    LENS— MICRO- 
SCOPE   METHOD. 

The  measurements  to  be  taken  are  the  same  as  in  the 
measurement  of  refractive  index  of  a  plate — namely,  the  real 
thickness,  and  the  apparent  thickness,  of  the  lens,  both  taken 
along  its  axis.  It  is  evident  that  the  formula  for  the  plate  will 
not  apply  here,  since  the  wave-front  in  the  air,  after  refraction 
at  the  upper  spherical  surface  of  the  lens,  will  differ  in  form 
from  that  due  to  refraction  at  the  plane  upper  surface  of  the 
plate.  By  means  of  the  known  relations  for  refraction  at  a 
single  spherical  surface,  show  that 

AB(r—Ab\  where  AB  =  real   thickness   of  lens 

n  =  Ab(   -AB\  ^4^  =  apparent    " 

r  =  radius  of  upper  surface. 

^  Set  the  lens,  resting  on  a  piece  of  plate  glass,  on  the  stage 
of  the  microscope,  with  its  axis  as  nearly  as  possible  in  the 
axis  of  the  microscope.  A  little  chalk  dust  rubbed  with  the 
finger  from  a  crayon  and  smeared  very  lightly  on  the  upper 
surfaces  of  the  lens  and  the  plate  glass,  will  provide  objects 
on  which  to  focus  the  microscope.  Focus  roughly  on  the 


74  WRITTEN    EXERCISE.       LENSES.  [43 

upper  surface  of  the  lens  by  sliding  the  tube  in  its  clamp,  then 
clamp  the  tube  and  focus  more  closely  by  means  of  the 
micrometer  screw,  which  moves  the  tube  vertically.  See 
that  the  tube  is  parallel  to  the  screw,  and  that  it  can  be 
moved  downward  through  a  distance  fully  equal  to  the  thick- 
ness of  the  lens. 

In  making  a  setting,  always  move  the  screw  in  the  same 
direction  to  the  desired  point;  and  if  that  point  be  accidentally 
passed,  turn  the  screw  backward  some  distance,  in  order  to 
approach  the  point  again  from  the  original  direction.  This 
precaution  is!  necessary  because  there  is  likely  to  be  some 
lost  motion  in  changing  the  direction  of  rotation  of  the  screw. 

Use  the  parallax  test,  as  far  as  possible,  to  decide  whether 
the  image  is  accurately  in  the  plane  of  the  cross-wires. 

The  pitch  of  the  screw  is  0.025  cm. 

After  practicing  until  settings  can  be  made  which  agree 
fairly  well,  let  each  observer  make  a  series  of  settings  on  (a) 
the  upper  surface  of  the  lens,  (&)  the  upper  surface  of  the  plate 
as  seen  through  the  lens,  and  (c)  the  upper  surface  of  the  plate 
after  removing  the  lens. 

From  these  find  AB  and  Ab,  and  calculate  n,  the  index  of 
refraction. 

Tabulate  all  the  indices  of  refraction  which  have  been 
found. 

43.  WRITTEN    EXERCISE.     LENSES. 

1.  Show  how  the  optical  bench,  as  used  in    Exps.  39,  40, 
corresponds  to  the  type  of  (a)  the  astronomical  telescope,  or 
(b)  the  compound  microscope,  according  to  the  position   of 
the  large  lens  with  reference  to  the  two  screens. 

2.  Taking  either  of  the  cases  in  (i),  trace  the  light  from  the 
object  which  falls  on  the  effective  aperture  of  the  large  lens, 
until  it  finally  forms  an  image  of  the  object  on  the  retina  of 
the  eye;    indicating  the   position  and   kind  of  image  formed 
by  refraction  at  each  lens  of  the  system. 


44]  DIFFRACTION    GRATING.  75 

3.  Describe   the    essential    improvements    in    the   working 
telescope  and  microscope  over  the  optical  bench  types. 

4.  Certain  approximations  are  made  in  deriving  the  funda- 
mental formulae  relating  to  lenses  which  you  have  used  in 
Exps.    39—42.      State   these   approximations,  and   show  that 
under  the  working  conditions,  their  use  is  justified. 

5.  Explain  the  theory  of  the  following  method  of  finding 
the  focal  length  of  a  diverging  (concave)  lens: — 

Threads  stretched  across  an  aperture  in  a  screen,  and 
brightly  illuminated,  form  the  object  o.  This  is  put  at  one 
end  of  an  optical  bench,  next  to  it  a  converging  lens,  then 
the  diverging  lens,  and  at  the  other  end  of  the  bench  a  plane 
mirror,  facing  the  object,  and  with  its  plane  at  right  angles 
to  the  axis  of  the  bench. 

The  lenses  are  now  so  adjusted  that  the  light  from  the 
object,  passing  through  the  lenses  to  the  mirror,  and  there 
reflected  back  so  as  to  again  pass  through  the  lenses  in  the 
reverse  direction,  forms  an  image  i  of  the  object  on  the 
screen  beside  the  aperture — that  is,  the  image  is  formed  in 
the  same  plane  as  the  object. 

If  now  the  diverg- 
ing lens  be  re- 
moved, the  con- 
verging lens  will 
form  an  image  i  of  the  object  at  some  point  beyond  the 
place  where  the  diverging  lens  was.  This  image  may  be 
located  by  any  of  the  usual  methods,  and  its  position  was 
the  principal  focus  of  the  diverging  lens  when  that  was  in 
position.  Then  of  course  the  distance  from  this  point  to  the 
point  where  the  lens  was,  is  its  focal  length. 

44.  DIFFRACTION    GRATING. 

A  monochromatic  light  is  required.  For  this  purpose 
use  a  Bunsen  burner  with  an  asbestos  cap  which  can  be 


76  DIFFRACTION    GRATING.  [44 

soaked  in  a  solution  of  sodium  chloride,  so  as  to  give  a 
strong  sodium  flame.  Set  in  front  of  this  the  brass  screen, 
with  its  plane  vertical  and  the  slits  horizontal.  In  use,  the 
light  is  allowed  to  shine  through  one  slit  on  each  side  of  the 
central  vertical  line  of  the  screen,  the  third  slit  being  covered. 

The  grating  is  to  be  mounted  in  a  slide  on  a  meter  rod, 
with  the  ruled  surface  toward  the  screen  and  the  lines  parallel 
to  the  slits.  The  rod  is  to  be  supported  by  a  clamp  so  that 
it  will  carry  the  grating  along  a  line  perpendicular  to  the 
screen  at  the  lower  slit.  On  looking  at  the  slits  through 
the  grating,  a  series  of  diffracted  images  of  each  slit  is  seen. 
The  two  sets  of  images  move  relatively  to  each  other  as  the 
grating  is  moved  back  and  forth;  and  a  position  of  the 
grating  may  be  found  for  which  one  of  the  images  in  one  set 
appears  to  form  a  continuous  line  with  one  in  the  other  set. 
If  the  lines  do  not  quite  meet,  it  is  evidence  that  the  lines  of 
the  grating  are  not  quite  parallel  to  the  slits. 

(a.)  Using  the  two  slits  which  are  closer  together,  set  the 
grating  so  that  the  first  image  of  one  slit  coincides,  in  the 
way  above  indicated,  with  the  first  image  of  the  other  slit, 
at  the  point  half-way  between  the  two  slits.  Make  a  number 
of  settings,  and  for  the  mean  of  these  settings  determine  the 
distance  of  the  ruled  surface  of  the  grating  from  the  screen. 

(&)  Repeat  (a),  using  the  second  diffracted  image  of  each 
slit,  instead  of  the  first. 

(<r.)  Repeat  (a)  and  (b)  for  the  pair  of  slits  which  are  wider 
apart. 

(d.)  Measure  the  distance  between  the  slits  with  the  y2  mm. 
glass  scale.  Determine  the  mean  distance  from  measure- 
ments on  the  edges,  taking  a  number  of  readings,  on  different 
parts  of  the  scale,  for  each  distance. 

(e.)  Assuming  the  wave-length  of  sodium  light  to  be 
0.0000589  cm.,  determine  the  distance  apart  of  the  lines  in 
the  grating.  Explain  fully  your  method  of  calculation,  and 
show  that  it  applies  to  this  case. 


673151          ^7 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 


